{"id":2815,"date":"2023-06-17T23:42:57","date_gmt":"2023-06-17T21:42:57","guid":{"rendered":"http:\/\/ff.mhrooz.xyz\/?p=2815"},"modified":"2023-06-17T23:43:00","modified_gmt":"2023-06-17T21:43:00","slug":"statistical_signal_processing_-_summary","status":"publish","type":"post","link":"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/06\/17\/statistical_signal_processing_-_summary\/","title":{"rendered":"Statistical Signal Processing &#8211; Summary"},"content":{"rendered":"\n<!doctype html>\n<html>\n<head>\n<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>\n\n<style type='text\/css'>html {overflow-x: initial !important;}:root { --bg-color: #ffffff; --text-color: #333333; --select-text-bg-color: #B5D6FC; --select-text-font-color: auto; --monospace: \"Lucida Console\",Consolas,\"Courier\",monospace; --title-bar-height: 20px; }\n.mac-os-11 { --title-bar-height: 28px; }\nhtml { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); 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{padding:0;}}\n<\/style><title>Statistical Signal Processing<\/title>\n<\/head>\n<body class='typora-export'><div class='typora-export-content'>\n<div id='write'  class=''><h1 id='i-parameter-estimation'><span>I. Parameter Estimation<\/span><\/h1><h2 id='1-statistical-modeling-and-quality-criteria'><span>1 Statistical Modeling and Quality Criteria<\/span><\/h2><ul><li><p><span>Statistical Estimation<\/span><\/p><\/li><\/ul><p><span>The goal is to determine the probability distribution of the random variables based on avaliable samples. \/ The stochastic model is a set of probability spaces and the task of statistical estimation is to select the most appropriate candidate based on the observed outcomes of a random experiment.<\/span><\/p><ul><li><p><span>Statistical Model<\/span><\/p><\/li><\/ul><p><span>In a statistical model, data is assumed to be generated by some underlying probability distribution, and the goal is to estimate the parameters of this distribution.<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.633ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 722 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-109-TEX-D-1D54F\" d=\"M22 666Q22 677 31 680T80 683H184H335Q346 675 346 667Q346 660 341 655Q335 648 315 648Q280 644 273 637Q273 630 300 583T356 492T386 448Q430 504 450 535T474 577T478 601Q478 620 469 634T444 648Q428 648 428 666Q428 678 436 680T488 683H559H630Q673 683 681 681T690 666Q690 648 673 648Q652 648 619 637Q571 615 517 550Q490 517 450 464T410 408Q415 399 501 273T617 106Q648 61 661 48T688 35Q705 35 705 16Q705 5 695 -1H539Q384 -1 379 3Q373 10 373 17Q373 27 380 31T408 35Q459 40 459 49Q459 59 418 129T335 259Q334 260 332 260Q328 260 273 197Q210 127 208 117Q199 104 199 82Q199 57 213 46T239 35Q247 35 252 29T257 15Q257 10 256 7T253 3T248 0L246 -1H28Q16 7 16 15T21 29T35 35Q61 35 117 88Q289 279 304 297Q307 303 255 377Q117 586 79 626Q60 648 39 648Q32 648 27 653T22 666ZM237 639V648H173Q113 647 113 646Q113 642 137 612Q186 546 302 373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-109-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{X}<\/script><span>: a set of all possible observations<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.382ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 611 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-103-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-103-TEX-D-1D53D\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}<\/script><span>: a subset of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.633ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 722 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-109-TEX-D-1D54F\" d=\"M22 666Q22 677 31 680T80 683H184H335Q346 675 346 667Q346 660 341 655Q335 648 315 648Q280 644 273 637Q273 630 300 583T356 492T386 448Q430 504 450 535T474 577T478 601Q478 620 469 634T444 648Q428 648 428 666Q428 678 436 680T488 683H559H630Q673 683 681 681T690 666Q690 648 673 648Q652 648 619 637Q571 615 517 550Q490 517 450 464T410 408Q415 399 501 273T617 106Q648 61 661 48T688 35Q705 35 705 16Q705 5 695 -1H539Q384 -1 379 3Q373 10 373 17Q373 27 380 31T408 35Q459 40 459 49Q459 59 418 129T335 259Q334 260 332 260Q328 260 273 197Q210 127 208 117Q199 104 199 82Q199 57 213 46T239 35Q247 35 252 29T257 15Q257 10 256 7T253 3T248 0L246 -1H28Q16 7 16 15T21 29T35 35Q61 35 117 88Q289 279 304 297Q307 303 255 377Q117 586 79 626Q60 648 39 648Q32 648 27 653T22 666ZM237 639V648H173Q113 647 113 646Q113 642 137 612Q186 546 302 373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-109-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{X}<\/script><span> that satisfies certain properties. Here, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.382ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 611 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-103-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-103-TEX-D-1D53D\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}<\/script><span> is the collection of events that can be assigned probabilities<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg 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749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"64\" xlink:href=\"#MJX-106-TEX-N-64\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D439\" xlink:href=\"#MJX-106-TEX-I-1D439\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1305,0)\"><use data-c=\"28\" xlink:href=\"#MJX-106-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1694,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-106-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1972,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-106-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2416.7,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-106-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2885.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-106-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">d<\/mi><\/mrow><mi>F<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathrm{d}F(\\cdot;\\theta)<\/script><span>: a set of optional probability measures that assign probabilities to events in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.382ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 611 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-103-TEX-D-1D53D\" d=\"M584 499Q569 490 566 490Q558 490 552 497T546 515Q546 535 533 559Q526 574 506 593T469 621Q415 648 326 648Q293 648 287 647T275 641Q264 630 263 617Q262 609 260 492V370L275 372Q323 376 350 392T393 441Q409 473 409 506Q409 529 427 529Q437 529 442 519Q444 511 444 362Q444 212 442 206Q436 197 426 197Q409 197 409 217Q409 265 375 299Q346 328 280 335H260V206Q260 70 262 63Q265 46 276 41T326 35Q362 35 366 28Q377 17 366 3L360 -1H24Q12 5 12 16Q12 35 51 35Q92 38 97 52Q102 60 102 341T97 632Q91 645 51 648Q12 648 12 666Q12 675 24 683H573Q576 678 584 670V499ZM137 341Q137 131 136 89T130 37Q129 36 129 35H182Q233 35 233 39Q226 54 225 92T224 346L226 623L231 635L235 648H129Q132 641 133 638T135 603T137 517T137 341ZM549 603V648H495L506 641Q531 621 533 619L549 603ZM409 317V395L400 386Q390 376 375 366L357 355L373 346Q394 331 397 328L409 317Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D53D\" xlink:href=\"#MJX-103-TEX-D-1D53D\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">F<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{F}<\/script><span>, they depend on parameter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>: selects the probability measure <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.409ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3274.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-106-TEX-N-64\" d=\"M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z\"><\/path><path id=\"MJX-106-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-106-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-106-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-106-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-106-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-106-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"64\" xlink:href=\"#MJX-106-TEX-N-64\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(556,0)\"><use data-c=\"1D439\" xlink:href=\"#MJX-106-TEX-I-1D439\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1305,0)\"><use data-c=\"28\" xlink:href=\"#MJX-106-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1694,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-106-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1972,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-106-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2416.7,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-106-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2885.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-106-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">d<\/mi><\/mrow><mi>F<\/mi><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathrm{d}F(\\cdot;\\theta)<\/script><span>. They are unknown and need to be estimated based on the available data.<\/span><\/p><ul><li><p><span>Statistics<\/span><\/p><\/li><\/ul><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.285ex\" height=\"1.95ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -704 4988.1 861.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-107-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-107-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-107-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-107-TEX-N-3A9\" d=\"M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z\"><\/path><path id=\"MJX-107-TEX-N-2192\" d=\"M56 237T56 250T70 270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-107-TEX-D-1D54F\" d=\"M22 666Q22 677 31 680T80 683H184H335Q346 675 346 667Q346 660 341 655Q335 648 315 648Q280 644 273 637Q273 630 300 583T356 492T386 448Q430 504 450 535T474 577T478 601Q478 620 469 634T444 648Q428 648 428 666Q428 678 436 680T488 683H559H630Q673 683 681 681T690 666Q690 648 673 648Q652 648 619 637Q571 615 517 550Q490 517 450 464T410 408Q415 399 501 273T617 106Q648 61 661 48T688 35Q705 35 705 16Q705 5 695 -1H539Q384 -1 379 3Q373 10 373 17Q373 27 380 31T408 35Q459 40 459 49Q459 59 418 129T335 259Q334 260 332 260Q328 260 273 197Q210 127 208 117Q199 104 199 82Q199 57 213 46T239 35Q247 35 252 29T257 15Q257 10 256 7T253 3T248 0L246 -1H28Q16 7 16 15T21 29T35 35Q61 35 117 88Q289 279 304 297Q307 303 255 377Q117 586 79 626Q60 648 39 648Q32 648 27 653T22 666ZM237 639V648H173Q113 647 113 646Q113 642 137 612Q186 546 302 373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-107-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-107-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1432.7,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-107-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1988.5,0)\"><use data-c=\"3A9\" xlink:href=\"#MJX-107-TEX-N-3A9\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2988.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-107-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4266.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-107-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>:<\/mo><mi mathvariant=\"normal\">\u03a9<\/mi><mo stretchy=\"false\">\u2192<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_i: \\Omega \\rightarrow \\mathbb{X}<\/script><span>, a random variable, also called a statistic, maps outcomes in the sample space <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.633ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -704 722 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-108-TEX-N-3A9\" d=\"M55 454Q55 503 75 546T127 617T197 665T272 695T337 704H352Q396 704 404 703Q527 687 596 615T666 454Q666 392 635 330T559 200T499 83V80H543Q589 81 600 83T617 93Q622 102 629 135T636 172L637 177H677V175L660 89Q645 3 644 2V0H552H488Q461 0 456 3T451 20Q451 89 499 235T548 455Q548 512 530 555T483 622T424 656T361 668Q332 668 303 658T243 626T193 560T174 456Q174 380 222 233T270 20Q270 7 263 0H77V2Q76 3 61 89L44 175V177H84L85 172Q85 171 88 155T96 119T104 93Q109 86 120 84T178 80H222V83Q206 132 162 199T87 329T55 454Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"3A9\" xlink:href=\"#MJX-108-TEX-N-3A9\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u03a9<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Omega<\/script><span> to values in set <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.633ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 722 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-109-TEX-D-1D54F\" d=\"M22 666Q22 677 31 680T80 683H184H335Q346 675 346 667Q346 660 341 655Q335 648 315 648Q280 644 273 637Q273 630 300 583T356 492T386 448Q430 504 450 535T474 577T478 601Q478 620 469 634T444 648Q428 648 428 666Q428 678 436 680T488 683H559H630Q673 683 681 681T690 666Q690 648 673 648Q652 648 619 637Q571 615 517 550Q490 517 450 464T410 408Q415 399 501 273T617 106Q648 61 661 48T688 35Q705 35 705 16Q705 5 695 -1H539Q384 -1 379 3Q373 10 373 17Q373 27 380 31T408 35Q459 40 459 49Q459 59 418 129T335 259Q334 260 332 260Q328 260 273 197Q210 127 208 117Q199 104 199 82Q199 57 213 46T239 35Q247 35 252 29T257 15Q257 10 256 7T253 3T248 0L246 -1H28Q16 7 16 15T21 29T35 35Q61 35 117 88Q289 279 304 297Q307 303 255 377Q117 586 79 626Q60 648 39 648Q32 648 27 653T22 666ZM237 639V648H173Q113 647 113 646Q113 642 137 612Q186 546 302 373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-109-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathbb{X}<\/script><span>, representing the values of interest that we want to study in our statistical analysis<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.433ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2843.5 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-112-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 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46T239 35Q247 35 252 29T257 15Q257 10 256 7T253 3T248 0L246 -1H28Q16 7 16 15T21 29T35 35Q61 35 117 88Q289 279 304 297Q307 303 255 377Q117 586 79 626Q60 648 39 648Q32 648 27 653T22 666ZM237 639V648H173Q113 647 113 646Q113 642 137 612Q186 546 302 373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-112-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-112-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1176.7,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-112-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2121.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-112-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_i\\in\\mathbb{X}<\/script><span>: actual values that we observe or measure in our study<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.392ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4593.1 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-111-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-111-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-111-TEX-D-1D54F\" d=\"M22 666Q22 677 31 680T80 683H184H335Q346 675 346 667Q346 660 341 655Q335 648 315 648Q280 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xlink:href=\"#MJX-111-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(981.8,0)\"><use data-c=\"3A\" xlink:href=\"#MJX-111-TEX-N-3A\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1537.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-111-TEX-D-1D54F\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2537.3,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-111-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3815.1,0)\"><use data-c=\"398\" xlink:href=\"#MJX-111-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><mo stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T:\\mathbb{X}\\rightarrow \\Theta<\/script><span>, a special type of statistic, called an estimator, maps observations <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.433ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2843.5 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-112-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-112-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-112-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 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373T453 139Q497 63 497 43Q497 39 495 35H559Q622 35 622 37Q622 38 583 94T486 233T373 399T277 552T237 639ZM553 637L566 648H504L508 637Q510 630 515 615V603L528 615Q529 616 539 625T553 637ZM170 46Q169 49 167 58T164 70V83L137 59L113 35H175Q175 38 170 46Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-112-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-112-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1176.7,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-112-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2121.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D54F\" xlink:href=\"#MJX-112-TEX-D-1D54F\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_i\\in\\mathbb{X}<\/script><span> to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.587ex\" height=\"2.518ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1073 2469.6 1113\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.09ex;\"><defs><path id=\"MJX-113-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-113-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-113-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-113-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-113-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(317.8,279) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-113-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(746.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-113-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1691.6,0)\"><use data-c=\"398\" xlink:href=\"#MJX-113-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>\u2208<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{\\theta}\\in\\Theta<\/script><span> <\/span><\/p><ul><li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.78ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2112.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-114-TEX-I-1D443\" d=\"M287 628Q287 635 230 637Q206 637 199 638T192 648Q192 649 194 659Q200 679 203 681T397 683Q587 682 600 680Q664 669 707 631T751 530Q751 453 685 389Q616 321 507 303Q500 302 402 301H307L277 182Q247 66 247 59Q247 55 248 54T255 50T272 48T305 46H336Q342 37 342 35Q342 19 335 5Q330 0 319 0Q316 0 282 1T182 2Q120 2 87 2T51 1Q33 1 33 11Q33 13 36 25Q40 41 44 43T67 46Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628ZM645 554Q645 567 643 575T634 597T609 619T560 635Q553 636 480 637Q463 637 445 637T416 636T404 636Q391 635 386 627Q384 621 367 550T332 412T314 344Q314 342 395 342H407H430Q542 342 590 392Q617 419 631 471T645 554Z\"><\/path><path id=\"MJX-114-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-114-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-114-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-114-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D443\" xlink:href=\"#MJX-114-TEX-I-1D443\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-114-TEX-I-1D703\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1056.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-114-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1445.6,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-114-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1723.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-114-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>P<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">P_{\\theta}(\\cdot)<\/script><span> refers to the probability of an event according to the probability distribution selected by <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>. The same holds for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.495ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1986.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-116-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-116-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-116-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-116-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-116-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-116-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(771,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-116-TEX-I-1D703\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1152.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-116-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1430.6,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-116-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1708.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-116-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E_{\\theta}[\\cdot]<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.54ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2890.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-117-TEX-N-56\" d=\"M114 620Q113 621 110 624T107 627T103 630T98 632T91 634T80 635T67 636T48 637H19V683H28Q46 680 152 680Q273 680 294 683H305V637H284Q223 634 223 620Q223 618 313 372T404 126L490 358Q575 588 575 597Q575 616 554 626T508 637H503V683H512Q527 680 627 680Q718 680 724 683H730V637H723Q648 637 627 596Q627 595 515 291T401 -14Q396 -22 382 -22H374H367Q353 -22 348 -14Q346 -12 231 303Q114 617 114 620Z\"><\/path><path id=\"MJX-117-TEX-N-61\" d=\"M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z\"><\/path><path id=\"MJX-117-TEX-N-72\" d=\"M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z\"><\/path><path id=\"MJX-117-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-117-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-117-TEX-N-22C5\" d=\"M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z\"><\/path><path id=\"MJX-117-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mtext\"><use data-c=\"56\" xlink:href=\"#MJX-117-TEX-N-56\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-117-TEX-N-61\" transform=\"translate(750,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-117-TEX-N-72\" transform=\"translate(1250,0)\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(1675,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-117-TEX-I-1D703\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2056.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-117-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2334.6,0)\"><use data-c=\"22C5\" xlink:href=\"#MJX-117-TEX-N-22C5\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2612.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-117-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mtext>Var<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\text{Var}_{\\theta}[\\cdot]<\/script><span>.<\/span><\/p><\/li><\/ul><h3 id='11-introductory-example-estimating-upper-bound-\\n\u03b8'><span>1.1 Introductory Example: Estimating upper bound <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><\/h3><p><span>Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> i.i.d. statistics <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.383ex\" height=\"1.984ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 5031.5 877\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-218-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-218-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-218-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-218-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-218-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-218-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-218-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-218-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1709.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-218-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3047.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-218-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3492.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-218-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-218-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_1,\\dots,X_N<\/script><span> of a uniformly distributed random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.458ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5064.2 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-450-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-450-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-450-TEX-C-55\" d=\"M8 592Q8 616 70 649T193 683Q246 683 246 631Q246 587 205 492T124 297T83 143Q83 101 100 75T154 48Q202 48 287 135T450 342T560 553Q589 635 593 640Q603 656 626 668T669 683H670Q687 683 687 672T670 616T617 463T547 220Q525 137 521 68Q521 54 522 50T533 42L543 47Q573 61 588 61Q604 61 604 47Q599 16 506 -22Q486 -28 468 -28T436 -18T421 18Q421 92 468 258Q468 259 467 257T459 248Q426 206 391 167T303 81T194 6T83 -22Q66 -22 58 -20Q25 -11 4 19T-17 99Q-17 146 8 220T64 358T120 488T146 586Q146 604 141 608T123 613H120Q99 613 72 597T25 580Q8 580 8 592Z\"><\/path><path id=\"MJX-450-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-450-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-450-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-450-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-450-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-450-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1129.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-450-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2185.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"55\" xlink:href=\"#MJX-450-TEX-C-55\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2872.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-450-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3261.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-450-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3761.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-450-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4206.2,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-450-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4675.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-450-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">U<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X\\sim \\mathcal U (0,\\theta)<\/script><span> where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is deterministic but unknown. We have that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"27.863ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 12315.3 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-125-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-125-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-125-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-125-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-125-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-125-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-125-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-125-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-125-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-125-TEX-I-1D449\" d=\"M52 648Q52 670 65 683H76Q118 680 181 680Q299 680 320 683H330Q336 677 336 674T334 656Q329 641 325 637H304Q282 635 274 635Q245 630 242 620Q242 618 271 369T301 118L374 235Q447 352 520 471T595 594Q599 601 599 609Q599 633 555 637Q537 637 537 648Q537 649 539 661Q542 675 545 679T558 683Q560 683 570 683T604 682T668 681Q737 681 755 683H762Q769 676 769 672Q769 655 760 640Q757 637 743 637Q730 636 719 635T698 630T682 623T670 615T660 608T652 599T645 592L452 282Q272 -9 266 -16Q263 -18 259 -21L241 -22H234Q216 -22 216 -15Q213 -9 177 305Q139 623 138 626Q133 637 76 637H59Q52 642 52 648Z\"><\/path><path id=\"MJX-125-TEX-I-1D44E\" d=\"M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-125-TEX-I-1D45F\" d=\"M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q161 442 183 430T214 408T225 388Q227 382 228 382T236 389Q284 441 347 441H350Q398 441 422 400Q430 381 430 363Q430 333 417 315T391 292T366 288Q346 288 334 299T322 328Q322 376 378 392Q356 405 342 405Q286 405 239 331Q229 315 224 298T190 165Q156 25 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-125-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" 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data-mml-node=\"mn\" transform=\"translate(4474.6,0)\"><use data-c=\"32\" xlink:href=\"#MJX-125-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4974.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-125-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5419.2,0)\"><use data-c=\"1D449\" xlink:href=\"#MJX-125-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6188.2,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-125-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6717.2,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-125-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7168.2,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-125-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7446.2,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-125-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8298.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-125-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8854,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-125-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(9909.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-125-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(502,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-125-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10815.3,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-125-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(11315.3,0)\"><use data-c=\"31\" xlink:href=\"#MJX-125-TEX-N-31\"><\/use><use data-c=\"32\" xlink:href=\"#MJX-125-TEX-N-32\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mn>2<\/mn><mo>,<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mn>12<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]=\\theta\/2,Var[X]=\\theta^2\/12<\/script><span>.<\/span><\/p><h4 id='solution-1-use-\\ne\\n\\nx\\n\\n\\n\u03b8\\n\\n2'><span>Solution 1: Use <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.255ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4974.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-473-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-473-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-473-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-473-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-473-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-473-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-473-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-473-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-473-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-473-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-473-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1894,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-473-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2449.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-473-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3505.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-473-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3974.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-473-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(4474.6,0)\"><use data-c=\"32\" xlink:href=\"#MJX-473-TEX-N-32\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mn>2<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]=\\theta\/2<\/script><\/h4><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n453\" cid=\"n453\" mdtype=\"math_block\" 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displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo>:<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">\u21a6<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mn>1<\/mn><\/msub><mo>=<\/mo><mn>2<\/mn><mo>\u22c5<\/mo><mfrac><mn>1<\/mn><mi>N<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>X<\/mi><mi>i<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='solution-2-intuition-\\n\u03b8\\n\\n\\nmax\\ni\\n\\n1\\n\\n\\n\\nn\\n\\nx\\ni\\n'><span>Solution 2: Intuition <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"20.133ex\" height=\"3.077ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1073 8898.9 1360.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-1180-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-1180-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-1180-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-1180-TEX-N-6D\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q351 442 364 440T387 434T406 426T421 417T432 406T441 395T448 384T452 374T455 366L457 361L460 365Q463 369 466 373T475 384T488 397T503 410T523 422T546 432T572 439T603 442Q729 442 740 329Q741 322 741 190V104Q741 66 743 59T754 49Q775 46 803 46H819V0H811L788 1Q764 2 737 2T699 3Q596 3 587 0H579V46H595Q656 46 656 62Q657 64 657 200Q656 335 655 343Q649 371 635 385T611 402T585 404Q540 404 506 370Q479 343 472 315T464 232V168V108Q464 78 465 68T468 55T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 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431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z\"><\/path><path id=\"MJX-1180-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 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10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-1180-TEX-N-7D\" d=\"M65 731Q65 745 68 747T88 750Q171 750 216 725T279 670Q288 649 289 635T291 501Q292 362 293 357Q306 312 345 291T417 269Q428 269 431 266T434 250T431 234T417 231Q380 231 345 210T298 157Q293 143 292 121T291 -28V-79Q291 -134 285 -156T256 -198Q202 -250 89 -250Q71 -250 68 -247T65 -230Q65 -224 65 -223T66 -218T69 -214T77 -213Q91 -213 108 -210T146 -200T183 -177T207 -139Q208 -134 209 3L210 139Q223 196 280 230Q315 247 330 250Q305 257 280 270Q225 304 212 352L210 362L209 498Q208 635 207 640Q195 680 154 696T77 713Q68 713 67 716T65 731Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" 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data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-1180-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8398.9,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-1180-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>N<\/mi><\/mrow><\/munder><mo fence=\"false\" stretchy=\"false\">{<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{\\theta}=\\max_{i=1,\\dots,N}\\{X_i\\}<\/script><\/h4><h3 id='12-consistency-and-unbiasedness'><span>1.2 Consistency and Unbiasedness<\/span><\/h3><h4 id='consistent-estimator-\\nt\\n-\\nlim\\nn\\n\u2192\\n\u221e\\nt\\n\\nx\\n1\\n\\n\\n\\nx\\nn\\n\\n\u2192\\n\u03b8'><span>Consistent estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 704 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-166-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 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data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-128-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5782.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-128-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6227,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-128-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7565.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-128-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8010.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-128-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-128-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9293.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-128-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9960,0)\"><use data-c=\"2192\" xlink:href=\"#MJX-128-TEX-N-2192\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11237.8,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-128-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">lim<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>N<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/munder><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">\u2192<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lim_{N\\rightarrow\\infin}T(x_1,\\dots,x_N)\\rightarrow \\theta<\/script><\/h4><p><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.309ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1020.6 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1780-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1780-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1780-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1780-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_1<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.309ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1020.6 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1793-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1793-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1793-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1793-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_2<\/script><span> are both consistent.<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.309ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1020.6 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1780-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1780-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1780-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1780-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_1<\/script><span>:<\/span><\/p><p><span>Using Chebyshev inequality, we derive the law of large numbers,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n816\" cid=\"n816\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"63.548ex\" height=\"29.204ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6704.1 28088.2 12908.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -14.036ex;\"><defs><path id=\"MJX-5597-TEX-N-50\" d=\"M130 622Q123 629 119 631T103 634T60 637H27V683H214Q237 683 276 683T331 684Q419 684 471 671T567 616Q624 563 624 489Q624 421 573 372T451 307Q429 302 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transform=\"translate(9887,0)\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mi><\/mi><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2265<\/mo><msqrt><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/msqrt><mo>\u22c5<\/mo><mi>\u03f5<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mfrac><mn>1<\/mn><msup><mi>\u03f5<\/mi><mn>2<\/mn><\/msup><\/mfrac><mtext>&nbsp;(Chebyshev\u2019s inequality)<\/mtext><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo>=<\/mo><mfrac><mn>1<\/mn><mi>n<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>n<\/mi><\/munderover><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mfrac><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>n<\/mi><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mi><\/mi><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2265<\/mo><msqrt><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><\/msqrt><mo>\u22c5<\/mo><mi>\u03f5<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mfrac><mn>1<\/mn><msup><mi>\u03f5<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mi><\/mi><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2265<\/mo><msqrt><mfrac><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/mrow><mi>n<\/mi><\/mfrac><\/msqrt><mo>\u22c5<\/mo><mi>\u03f5<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mfrac><mn>1<\/mn><msup><mi>\u03f5<\/mi><mn>2<\/mn><\/msup><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><\/mtd><mtd><mi><\/mi><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">Pr<\/mo><mo stretchy=\"false\">(<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>X<\/mi><mo stretchy=\"false\">\u00af<\/mo><\/mover><\/mrow><mi>n<\/mi><\/msub><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo>\u2265<\/mo><mi>\u03f5<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2264<\/mo><mfrac><mrow data-mjx-texclass=\"ORD\"><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mi>n<\/mi><msup><mi>\u03f5<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mtext>&nbsp;(Law of large numbers)<\/mtext><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Using law of large numbers, we get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n818\" cid=\"n818\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"36.002ex\" height=\"4.957ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1345.5 15912.8 2190.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.913ex;\"><defs><path id=\"MJX-5598-TEX-N-50\" d=\"M130 622Q123 629 119 631T103 634T60 637H27V683H214Q237 683 276 683T331 684Q419 684 471 671T567 616Q624 563 624 489Q624 421 573 372T451 307Q429 302 328 301H234V181Q234 62 237 58Q245 47 304 46H337V0H326Q305 3 182 3Q47 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM507 488Q507 514 506 528T500 564T483 597T450 620T397 635Q385 637 307 637H286Q237 637 234 628Q231 624 231 483V342H302H339Q390 342 423 349T481 382Q507 411 507 488Z\"><\/path><path id=\"MJX-5598-TEX-N-72\" d=\"M36 46H50Q89 46 97 60V68Q97 77 97 91T98 122T98 161T98 203Q98 234 98 269T98 328L97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 60 434T96 436Q112 437 131 438T160 441T171 442H174V373Q213 441 271 441H277Q322 441 343 419T364 373Q364 352 351 337T313 322Q288 322 276 338T263 372Q263 381 265 388T270 400T273 405Q271 407 250 401Q234 393 226 386Q179 341 179 207V154Q179 141 179 127T179 101T180 81T180 66V61Q181 59 183 57T188 54T193 51T200 49T207 48T216 47T225 47T235 46T245 46H276V0H267Q249 3 140 3Q37 3 28 0H20V46H36Z\"><\/path><path id=\"MJX-5598-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5598-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-5598-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-5598-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-5598-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5598-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5598-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-5598-TEX-N-2265\" d=\"M83 616Q83 624 89 630T99 636Q107 636 253 568T543 431T687 361Q694 356 694 346T687 331Q685 329 395 192L107 56H101Q83 58 83 76Q83 77 83 79Q82 86 98 95Q117 105 248 167Q326 204 378 228L626 346L360 472Q291 505 200 548Q112 589 98 597T83 616ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-5598-TEX-I-1D716\" d=\"M227 -11Q149 -11 95 41T40 174Q40 262 87 322Q121 367 173 396T287 430Q289 431 329 431H367Q382 426 382 411Q382 385 341 385H325H312Q191 385 154 277L150 265H327Q340 256 340 246Q340 228 320 219H138V217Q128 187 128 143Q128 77 160 52T231 26Q258 26 284 36T326 57T343 68Q350 68 354 58T358 39Q358 36 357 35Q354 31 337 21T289 0T227 -11Z\"><\/path><path id=\"MJX-5598-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5598-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-5598-TEX-N-56\" d=\"M114 620Q113 621 110 624T107 627T103 630T98 632T91 634T80 635T67 636T48 637H19V683H28Q46 680 152 680Q273 680 294 683H305V637H284Q223 634 223 620Q223 618 313 372T404 126L490 358Q575 588 575 597Q575 616 554 626T508 637H503V683H512Q527 680 627 680Q718 680 724 683H730V637H723Q648 637 627 596Q627 595 515 291T401 -14Q396 -22 382 -22H374H367Q353 -22 348 -14Q346 -12 231 303Q114 617 114 620Z\"><\/path><path id=\"MJX-5598-TEX-N-61\" d=\"M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z\"><\/path><path id=\"MJX-5598-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5598-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 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270H835Q719 357 692 493Q692 494 692 496T691 499Q691 511 708 511H711Q720 511 723 510T729 506T732 497T735 481T743 456Q765 389 816 336T935 261Q944 258 944 250Q944 244 939 241T915 231T877 212Q836 186 806 152T761 85T740 35T732 4Q730 -6 727 -8T711 -11Q691 -11 691 0Q691 7 696 25Q728 151 835 230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5598-TEX-N-221E\" d=\"M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z\"><\/path><path id=\"MJX-5598-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 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data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><mfrac><mrow><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mrow><mi>\u03b8<\/mi><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mrow><mi>\u03b8<\/mi><mo>\u2212<\/mo><mi>\u03f5<\/mi><\/mrow><mi>\u03b8<\/mi><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mi>N<\/mi><\/msup><mover><mo stretchy=\"false\">\u27f6<\/mo><mrow><mi>N<\/mi><mo stretchy=\"false\">\u2192<\/mo><mi mathvariant=\"normal\">\u221e<\/mi><\/mrow><\/mover><mn>0<\/mn><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='unbiased-estimator-\\nt\\n-\\ne\\n\\nt\\n\\nx\\n1\\n\\n\\n\\nx\\nn\\n\\n\\n\\n\u03b8'><span>Unbiased estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 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xlink:href=\"#MJX-1793-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_2<\/script><span> is asymptorically unbiased:<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n468\" cid=\"n468\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 39.142ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.653ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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xlink:href=\"#MJX-5601-TEX-I-1D709\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8550.9,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-5601-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8995.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5601-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9464.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5601-TEX-N-29\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9853.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"64\" xlink:href=\"#MJX-5601-TEX-N-64\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(10409.6,0)\"><use data-c=\"1D709\" xlink:href=\"#MJX-5601-TEX-I-1D709\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11125.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5601-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(12181.1,0)\"><g data-mml-node=\"mi\" transform=\"translate(1081.2,676)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5601-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-686)\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-5601-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1110.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5601-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2110.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5601-TEX-N-31\"><\/use><\/g><\/g><rect width=\"2810.4\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(15231.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5601-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1499.2 1 2498.5\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-87.3)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msubsup><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mn>0<\/mn><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msubsup><mi>\u03be<\/mi><mo>\u22c5<\/mo><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>T<\/mi><mn>2<\/mn><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03be<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">d<\/mi><\/mrow><mi>\u03be<\/mi><mo>=<\/mo><mfrac><mi>N<\/mi><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mi>\u03b8<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><strong><span>Proof.<\/span><\/strong><span> For <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.217ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 4074.1 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-137-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-137-TEX-N-2264\" d=\"M674 636Q682 636 688 630T694 615T687 601Q686 600 417 472L151 346L399 228Q687 92 691 87Q694 81 694 76Q694 58 676 56H670L382 192Q92 329 90 331Q83 336 83 348Q84 359 96 365Q104 369 382 500T665 634Q669 636 674 636ZM84 -118Q84 -108 99 -98H678Q694 -104 694 -118Q694 -130 679 -138H98Q84 -131 84 -118Z\"><\/path><path id=\"MJX-137-TEX-I-1D709\" d=\"M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 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data-mml-node=\"mi\" transform=\"translate(1833.6,0)\"><use data-c=\"1D709\" xlink:href=\"#MJX-137-TEX-I-1D709\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2549.3,0)\"><use data-c=\"2264\" xlink:href=\"#MJX-137-TEX-N-2264\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3605.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-137-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>0<\/mn><mo>\u2264<\/mo><mi>\u03be<\/mi><mo>\u2264<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">0\\leq \\xi\\leq\\theta<\/script><span>,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n470\" cid=\"n470\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container 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mathvariant=\"normal\">d<\/mi><\/mrow><mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">d<\/mi><\/mrow><mi>\u03be<\/mi><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mfrac><mi>\u03be<\/mi><mi>\u03b8<\/mi><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mi>N<\/mi><\/msup><mo>=<\/mo><mi>N<\/mi><mo>\u22c5<\/mo><mfrac><msup><mi>\u03be<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>N<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mi>\u03b8<\/mi><mi>N<\/mi><\/msup><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.347ex\" height=\"2.3ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 1037.5 1016.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.583ex;\"><defs><path id=\"MJX-138-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 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113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-138-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(793,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-138-TEX-V-2032\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-257.5) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-138-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>T<\/mi><mn>2<\/mn><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_2'<\/script><span> is unbiased:<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n832\" cid=\"n832\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 18.195ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.627ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.748ex; min-width: 18.195ex;\"><defs><path id=\"MJX-5603-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 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137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5603-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5603-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msubsup><mi>T<\/mi><mn>2<\/mn><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msubsup><mo>=<\/mo><mfrac><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/mfrac><msub><mi>T<\/mi><mn>2<\/mn><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-06-17 14.43.31.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-06-17 14.43.31\"><\/p><h3 id='13-variance-\\nvar\\n\\nt\\n\\n\\ne\\n\\n\\nt\\n\u2212\\ne\\n\\nt\\n\\n\\n2\\n'><span>1.3 Variance: <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" 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id='14-mean-squared-errormse-\\ne\\n\\n\u03b5\\n2\\n\\nt\\n\\n\\n\\ne\\n\\n\\nt\\n\u2212\\n\u03b8\\n\\n2\\n'><span>1.4 Mean Squared Error(MSE): <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"22.552ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 9968.1 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-2227-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 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343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-2227-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-2227-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-2227-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-2227-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 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transform=\"translate(499,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-2227-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1944.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2227-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2333.6,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-2227-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3037.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2227-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3426.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-2227-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3982.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2227-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5038.1,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-2227-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5802.1,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-2227-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6080.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2227-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6469.1,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-2227-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7395.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-2227-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8395.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-2227-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(8864.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-2227-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(422,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-2227-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9690.1,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-2227-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>\u03b5<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[\\varepsilon^2(T)]=E[(T-\\theta)^2]<\/script><\/h3><p><span>An extension of the variance is the MSE(mean squared error), where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is the true parameter.<\/span><\/p><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.309ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1020.6 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-1780-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1780-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1780-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1780-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mn>1<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_1<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.347ex\" height=\"2.3ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 1037.5 1016.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.583ex;\"><defs><path id=\"MJX-138-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-138-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-138-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-138-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(793,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-138-TEX-V-2032\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-257.5) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-138-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>T<\/mi><mn>2<\/mn><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_2'<\/script><span> are unbiased: <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.724ex\" height=\"2.3ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -759 7834.1 1016.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.583ex;\"><defs><path id=\"MJX-1783-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-1783-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-1783-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-1783-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-1783-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-1783-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-1783-TEX-V-2032\" d=\"M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z\"><\/path><path id=\"MJX-1783-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-1783-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-1783-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-1783-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1042,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1783-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-1783-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2062.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1783-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2618.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1783-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3674.1,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-1783-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4438.1,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-1783-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(4716.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-1783-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(793,363) scale(0.707)\"><use data-c=\"2032\" xlink:href=\"#MJX-1783-TEX-V-2032\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-257.5) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-1783-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5753.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-1783-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6309.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-1783-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7365.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-1783-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mn>1<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msubsup><mi>T<\/mi><mn>2<\/mn><mo data-mjx-alternate=\"1\">\u2032<\/mo><\/msubsup><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[T_1]=E[T_2']=\\theta<\/script><span>, therefore,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n478\" cid=\"n478\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" 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stretchy=\"false\">]<\/mo><mi>\u03b8<\/mi><mo>+<\/mo><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mn>2<\/mn><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/IMG_D15E3BA67391-1.jpeg\" referrerpolicy=\"no-referrer\" alt=\"IMG_D15E3BA67391-1\"><\/p><h3 id='15-biasvariance-trade-off'><span>1.5 Bias\/Variance Trade-Off<\/span><\/h3><p><span>MSE of an estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" 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xlink:href=\"#MJX-166-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T<\/script><span> is defined as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n482\" cid=\"n482\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 26.172ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.565ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.717ex; min-width: 26.172ex;\"><defs><path id=\"MJX-5607-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-5607-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-5607-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z\"><\/path><path id=\"MJX-5607-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5607-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 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132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -817)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 817) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4984.1 -817 1 1133.9\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-67)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-5607-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5607-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1042,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D700\" xlink:href=\"#MJX-5607-TEX-I-1D700\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(499,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5607-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1944.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5607-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2333.6,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-5607-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3037.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5607-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3426.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5607-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3982.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5607-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5038.1,0)\"><use data-c=\"1D438\" xlink:href=\"#MJX-5607-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5802.1,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5607-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6080.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5607-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6469.1,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-5607-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7395.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5607-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8395.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5607-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(8864.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-5607-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(422,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5607-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9690.1,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5607-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -817 1 1133.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-67)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>\u03b5<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>and can be decomposed into its bias and variance<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n484\" cid=\"n484\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"39.99ex\" height=\"5.81ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1533.9 17675.5 2567.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.339ex;\"><defs><path id=\"MJX-5608-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 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429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-143-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(640,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-143-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(617,-150) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-143-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b1<\/mi><msub><mi>T<\/mi><mn>2<\/mn><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\alpha T_2<\/script><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n488\" cid=\"n488\" mdtype=\"math_block\" 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xlink:href=\"#MJX-5609-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5609-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1666,0)\"><use data-c=\"32\" xlink:href=\"#MJX-5609-TEX-N-32\"><\/use><\/g><\/g><rect width=\"1731.6\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><rect width=\"4282\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>\u03b1<\/mi><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msup><mi>\u03b1<\/mi><mn>2<\/mn><\/msup><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mfrac><mrow><mi>N<\/mi><msup><mi>\u03b1<\/mi><mn>2<\/mn><\/msup><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><mi>B<\/mi><mi>i<\/mi><mi>a<\/mi><mi>s<\/mi><mo stretchy=\"false\">[<\/mo><mi>\u03b1<\/mi><msub><mi>T<\/mi><mn>2<\/mn><\/msub><msup><mo stretchy=\"false\">]<\/mo><mn>2<\/mn><\/msup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>\u03b1<\/mi><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mi>\u03b1<\/mi><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mn>2<\/mn><\/msub><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mrow><mi>\u03b1<\/mi><mi>N<\/mi><\/mrow><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><mo>\u2212<\/mo><mn>1<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b1<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mi>B<\/mi><mi>i<\/mi><mi>a<\/mi><mi>s<\/mi><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/mfrac><\/mtd><\/mtr><mtr><mtd><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b1<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mi>M<\/mi><mi>S<\/mi><mi>E<\/mi><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><mo>+<\/mo><mfrac><mn>1<\/mn><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Therefore, an <\/span><strong><span>unbiased<\/span><\/strong><span> estimator is not necessarily the <\/span><strong><span>optimal<\/span><\/strong><span> estimator, but for large <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span>, it is.<\/span><\/p><h2 id='2-maximum-likelihood-estimation'><span>2 Maximum Likelihood Estimation<\/span><\/h2><h3 id='21-maximum-likelihood-principle'><span>2.1 Maximum Likelihood Principle<\/span><\/h3><p><span>The maximum likelihood principle suggests to select a candidate probability measure such that the <\/span><strong><span>observed outcomes<\/span><\/strong><span> of the experiment <\/span><strong><span>become most probable<\/span><\/strong><span>. A maximum likelihood estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.28ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1891.7 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-157-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-157-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-157-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-157-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D440\" xlink:href=\"#MJX-157-TEX-I-1D440\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1051,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-157-TEX-I-1D43F\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_{ML}<\/script><span> picks the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> that maximizes the likelihood function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"15.216ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" 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xlink:href=\"#MJX-2077-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3861.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-2077-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4306.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-2077-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-2077-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5589.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-2077-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5867.5,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-2077-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6336.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2077-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L(x_1,\\dots,x_N|\\theta)<\/script><span>.<\/span><\/p><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-05-16 22.35.39.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-05-16 22.35.39\"><\/p><p><span>The likelihood function depends on the statistical model, assuming all observations are iid, we obtain,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n495\" cid=\"n495\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div 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data-mml-node=\"msub\" transform=\"translate(22463.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5610-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(23362.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5610-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(23640.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5610-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(24109.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5610-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1633)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtext\"><use data-c=\"63\" xlink:href=\"#MJX-5610-TEX-N-63\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5610-TEX-N-6F\" transform=\"translate(444,0)\"><\/use><use data-c=\"6E\" 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data-c=\"56\" xlink:href=\"#MJX-5610-TEX-N-56\" transform=\"translate(5993,0)\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-5610-TEX-N-2E\" transform=\"translate(6743,0)\"><\/use><use data-c=\"3A\" xlink:href=\"#MJX-5610-TEX-N-3A\" transform=\"translate(7021,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-5610-TEX-N-A0\" transform=\"translate(7299,0)\"><\/use><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(7549,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-5610-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(681,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5610-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1070,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5610-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2078.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2523.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5610-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3861.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4306.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5610-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5589.5,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5610-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5867.5,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5610-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6336.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5610-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7003.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5610-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(8059,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-5610-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(523,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5610-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5610-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1264.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1542.6,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5610-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2714.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2992.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5610-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5610-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11836.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5610-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(12225.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5610-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13233.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13678.5,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5610-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15017.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5610-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(15461.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5610-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(16744.7,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5610-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17022.7,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5610-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17491.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5610-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18158.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5610-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"munderover\" transform=\"translate(19214.3,0)\"><g data-mml-node=\"mo\"><use data-c=\"220F\" xlink:href=\"#MJX-5610-TEX-LO-220F\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(65.2,-1087.9) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-5610-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5610-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1123,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5610-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(325,1150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5610-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(20658.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-5610-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(523,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5610-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5610-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(22048.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5610-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(22437.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5610-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5610-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(23336.6,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5610-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(23614.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5610-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(24083.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5610-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mtext>discrete R.V.:&nbsp;<\/mtext><\/mtd><mtd><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u220f<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mtext>continuous R.V.:&nbsp;<\/mtext><\/mtd><mtd><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u220f<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Normally, we use log-likelihood function,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n497\" cid=\"n497\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"49.936ex\" height=\"14.158ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3378.9 22071.8 6257.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -6.513ex;\"><defs><path id=\"MJX-5611-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 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data-mml-node=\"mo\" transform=\"translate(11105.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5611-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>:<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">\u21a6<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u220f<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>In the slides, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.662ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2944.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-2187-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-2187-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-2187-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-2187-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-2187-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-2187-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43F\" xlink:href=\"#MJX-2187-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(681,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2187-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1070,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-2187-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1642,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-2187-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2086.7,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-2187-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2555.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2187-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L(x;\\theta)<\/script><span> is used rather than <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.285ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2778 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-2189-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-2189-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-2189-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-2189-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-2189-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-2189-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 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transform=\"translate(2389,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2189-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">L(x|\\theta)<\/script><span>.<\/span><\/p><\/li><\/ul><h3 id='22-parameter-estimation'><span>2.2 Parameter Estimation<\/span><\/h3><h4 id='channel-estimation'><span>Channel Estimation<\/span><\/h4><p><span>Consider an AWGN channel <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.498ex\" height=\"2.034ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 5082 899\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-147-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-147-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-147-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-147-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-147-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-147-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-147-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(767.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-147-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1823.6,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-147-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2399.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-147-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3193.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-147-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4194,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-147-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>h<\/mi><mi>x<\/mi><mo>+<\/mo><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y=hx+N<\/script><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.418ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 5930.8 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-148-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-148-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-148-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-148-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-148-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-148-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-148-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-148-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-148-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-148-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1165.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-148-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2221.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-148-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3200.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-148-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3589.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-148-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4089.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-148-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4534.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-148-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-148-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5541.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-148-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N\\sim \\mathcal{N}(0,\\sigma^2)<\/script><span>. We know that <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.368ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 6350.8 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-149-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-149-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-149-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-149-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-149-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-149-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><path id=\"MJX-149-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-149-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-149-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-149-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-149-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1040.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-149-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2096.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-149-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3075.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-149-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3464.6,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-149-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4040.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-149-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4509.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-149-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4954.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-149-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-149-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5961.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-149-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>h<\/mi><mi>s<\/mi><mo>,<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y\\sim \\mathcal{N}(hs,\\sigma^2)<\/script><span>, and with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.381ex\" height=\"1.781ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2378.6 787\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-150-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-150-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-150-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-150-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(746.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-150-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1802.6,0)\"><use data-c=\"210E\" xlink:href=\"#MJX-150-TEX-I-210E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><mo>=<\/mo><mi>h<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta=h<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n501\" cid=\"n501\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 37.445ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.801ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.335ex; min-width: 37.445ex;\"><defs><path id=\"MJX-5612-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-5612-TEX-I-1D44C\" d=\"M66 637Q54 637 49 637T39 638T32 641T30 647T33 664T42 682Q44 683 56 683Q104 680 165 680Q288 680 306 683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-5612-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5612-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5612-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-5612-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-5612-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5612-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 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46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5612-TEX-I-1D70B\" d=\"M132 -11Q98 -11 98 22V33L111 61Q186 219 220 334L228 358H196Q158 358 142 355T103 336Q92 329 81 318T62 297T53 285Q51 284 38 284Q19 284 19 294Q19 300 38 329T93 391T164 429Q171 431 389 431Q549 431 553 430Q573 423 573 402Q573 371 541 360Q535 358 472 358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z\"><\/path><path id=\"MJX-5612-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-5612-TEX-N-65\" d=\"M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z\"><\/path><path id=\"MJX-5612-TEX-N-78\" d=\"M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z\"><\/path><path id=\"MJX-5612-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><path id=\"MJX-5612-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-5612-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5612-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 55Q108 52 115 47T146 34T201 27Q237 27 263 38T301 66T318 97T323 122Q323 150 302 164T254 181T195 196T148 231Q131 256 131 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -1532)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1532) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"7475.4 -1532 1 2563.9\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-12)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-5612-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-5612-TEX-I-1D44C\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1112.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5612-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1501.5,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-5612-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1991.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-5612-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2436.2,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5612-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2905.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5612-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3572,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5612-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(4627.7,0)\"><g data-mml-node=\"mn\" transform=\"translate(1217,676)\"><use data-c=\"31\" xlink:href=\"#MJX-5612-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-915)\"><g data-mml-node=\"msqrt\"><g transform=\"translate(853,0)\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-5612-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(500,0)\"><use data-c=\"1D70B\" xlink:href=\"#MJX-5612-TEX-I-1D70B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(0,95)\"><use data-c=\"221A\" xlink:href=\"#MJX-5612-TEX-N-221A\"><\/use><\/g><rect width=\"1070\" height=\"60\" x=\"853\" y=\"835\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(1923,0)\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5612-TEX-I-1D70E\"><\/use><\/g><\/g><rect width=\"2694\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(7561.7,0)\"><use data-c=\"65\" xlink:href=\"#MJX-5612-TEX-N-65\"><\/use><use data-c=\"78\" xlink:href=\"#MJX-5612-TEX-N-78\" transform=\"translate(444,0)\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-5612-TEX-N-70\" transform=\"translate(972,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9089.7,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-5612-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9089.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5612-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9478.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5612-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(10256.7,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,710)\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-5612-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(389,0)\"><use data-c=\"1D466\" xlink:href=\"#MJX-5612-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1101.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5612-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2101.4,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5612-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2570.4,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-5612-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3039.4,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-5612-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(422,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5612-TEX-N-32\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(1398.7,-719.9)\"><g data-mml-node=\"mn\"><use data-c=\"32\" xlink:href=\"#MJX-5612-TEX-N-32\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(500,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5612-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,289) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5612-TEX-N-32\"><\/use><\/g><\/g><\/g><rect width=\"4065\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mo\" transform=\"translate(14561.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5612-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1532 1 2563.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-12)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>f<\/mi><mi>Y<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><msqrt><mn>2<\/mn><mi>\u03c0<\/mi><\/msqrt><mi>\u03c3<\/mi><\/mrow><\/mfrac><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>\u2212<\/mo><mi>\u03b8<\/mi><mi>s<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mrow><mn>2<\/mn><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> observations <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-2129-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-2129-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y<\/script><span> and training signals <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.294ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 572 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-2131-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-2131-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>x<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x<\/script><span>, the maximum-likelihood estimation is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n503\" cid=\"n503\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"49.461ex\" height=\"28.082ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6456.1 21861.8 12412.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -13.475ex;\"><defs><path id=\"MJX-5613-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path 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683H316Q322 677 322 674T320 656Q316 643 310 637H298Q242 637 242 624Q242 619 292 477T343 333L346 336Q350 340 358 349T379 373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z\"><\/path><path id=\"MJX-151-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-151-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(614,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-151-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>Y<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y_i<\/script><span> are correlated or when <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> is non-Gaussian distributed.<\/span><\/p><\/li><\/ul><h4 id='introductory-example-estimating-upper-bound-\\n\u03b8'><span>Introductory Example: Estimating upper bound <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><\/h4><p><span>Suppose the distribution of observations is uniform, the likelihood function of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" 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stretchy=\"false\">}<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='n\\n-bernoulli-experiments'><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> Bernoulli Experiments<\/span><\/h4><p><span>Given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> observations, the likelihood function of success number is <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n514\" cid=\"n514\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 54.523ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.986ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.428ex; min-width: 54.523ex;\"><defs><path id=\"MJX-5616-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 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data-mml-node=\"mo\" transform=\"translate(11154.8,0)\"><use data-c=\"28\" xlink:href=\"#MJX-156-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(11543.8,0)\"><use data-c=\"31\" xlink:href=\"#MJX-156-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12266,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-156-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13266.2,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-156-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13735.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-156-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>N<\/mi><mi>\u03b8<\/mi><mo>,<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>N<\/mi><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]=N\\theta,Var[X]=N\\theta(1-\\theta)<\/script><span>.<\/span><\/p><p><span>The ML-estimator is obtained by<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n517\" cid=\"n517\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"28.3ex\" height=\"16.21ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3832.5 12508.8 7165\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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xlink:href=\"#MJX-5617-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5617-TEX-I-1D441\"><\/use><\/g><rect width=\"1088\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>=<\/mo><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mfrac><mi>d<\/mi><mrow><mi>d<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mi>x<\/mi><mi>\u03b8<\/mi><\/mfrac><mo>\u2212<\/mo><mfrac><mrow><mi>N<\/mi><mo>\u2212<\/mo><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mi>x<\/mi><mi>N<\/mi><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>In the following, we analyze the quality of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.28ex\" height=\"1.871ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1891.7 827\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-157-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 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data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mfrac><mi>X<\/mi><mi>N<\/mi><\/mfrac><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mfrac><mrow><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><mi>N<\/mi><\/mfrac><mo>=<\/mo><mi>\u03b8<\/mi><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;unbiased<\/mtext><\/mtd><\/mtr><mtr><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/mrow><msup><mi>N<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>=<\/mo><mfrac><mrow><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>\u03b5<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mfrac><mrow><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo 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stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mrow><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>\u03b8<\/mi><\/mrow><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>N<\/mi><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-05-09 22.13.24.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-05-09 22.13.24\"><\/p><h3 id='23-best-unbiased-estimator'><span>2.3 Best Unbiased Estimator<\/span><\/h3><p><span>ML estimators are not necessarily the best estimators. However, a wide class of estimators is defined by minimizing the MSE under an unbiasedness constraint.<\/span><\/p><p><span>We call an estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 704 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-166-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-166-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T<\/script><span> <\/span><strong><span>Best Unbiased Estimator<\/span><\/strong><span> if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.801ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 9636 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path 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xlink:href=\"#MJX-5621-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2264<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>S<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>for any alternative <\/span><strong><span>unbiased<\/span><\/strong><span> estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-272-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 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transform=\"translate(10839.5,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 1)\"><use data-c=\"222B\" xlink:href=\"#MJX-5622-TEX-LO-222B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(589,-896.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5622-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(12247.6,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-5622-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12928.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5622-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13317.6,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-5622-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13889.6,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-5622-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14334.2,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5622-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14803.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5622-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15192.2,0)\"><use data-c=\"1D451\" xlink:href=\"#MJX-5622-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15712.2,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-5622-TEX-I-1D465\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2219<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi mathvariant=\"normal\">\u2200<\/mi><mi>x<\/mi><mo>\u2208<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">X<\/mi><\/mrow><mo>,<\/mo><mi>\u03b8<\/mi><mo>\u2208<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2219<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;differentiable with respect to&nbsp;<\/mtext><mi>\u03b8<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u2219<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mfrac><mi>\u2202<\/mi><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><mo>=<\/mo><mfrac><mi>\u2202<\/mi><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>x<\/mi><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We define the <\/span><strong><span>score function<\/span><\/strong><span> as the <\/span><strong><span>slope<\/span><\/strong><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.931ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4389.3 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-161-TEX-N-6C\" d=\"M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z\"><\/path><path id=\"MJX-161-TEX-N-6F\" d=\"M28 214Q28 309 93 378T250 448Q340 448 405 380T471 215Q471 120 407 55T250 -10Q153 -10 91 57T28 214ZM250 30Q372 30 372 193V225V250Q372 272 371 288T364 326T348 362T317 390T268 410Q263 411 252 411Q222 411 195 399Q152 377 139 338T126 246V226Q126 130 145 91Q177 30 250 30Z\"><\/path><path id=\"MJX-161-TEX-N-67\" d=\"M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z\"><\/path><path id=\"MJX-161-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-161-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-161-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-161-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-161-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-161-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-161-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"6C\" xlink:href=\"#MJX-161-TEX-N-6C\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-161-TEX-N-6F\" transform=\"translate(278,0)\"><\/use><use data-c=\"67\" xlink:href=\"#MJX-161-TEX-N-67\" transform=\"translate(778,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1278,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-161-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1444.7,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-161-TEX-I-1D43F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2125.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-161-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2514.7,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-161-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3086.7,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-161-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3531.3,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-161-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4000.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-161-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\log L(x;\\theta)<\/script><span> with respect to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>:<\/span><\/p><div 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stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mrow><mfrac><mi>\u2202<\/mi><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='31-cramer-rao-lower-bound'><span>3.1 Cramer-Rao Lower Bound<\/span><\/h3><p><span>With <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"35.702ex\" height=\"2.852ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -899.6 15780.1 1260.7\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.817ex;\"><defs><path id=\"MJX-2174-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 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transform=\"translate(1384,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-5625-TEX-I-1D452\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1467.5 1 2434.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-7.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mover><mo>\u2265<\/mo><mrow><mi>a<\/mi><mi>p<\/mi><mi>p<\/mi><mi>e<\/mi><mi>n<\/mi><mi>d<\/mi><mi>i<\/mi><mi>x<\/mi><\/mrow><\/mover><mo stretchy=\"false\">(<\/mo><mfrac><mrow><mi>\u2202<\/mi><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mfrac><mn>1<\/mn><mrow><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><msub><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><mo>=<\/mo><msub><mi>\u03b8<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>t<\/mi><mi>r<\/mi><mi>u<\/mi><mi>e<\/mi><\/mrow><\/msub><\/mrow><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>If <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.593ex\" height=\"1.532ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 704 677\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-166-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-166-TEX-I-1D447\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T<\/script><span> is unbiased, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.345ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5456.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-2162-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 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xlink:href=\"#MJX-2162-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D447\" xlink:href=\"#MJX-2162-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1746,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2162-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2135,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-2162-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2987,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2162-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3376,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-2162-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3931.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-2162-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4987.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-2162-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[T(X)]=\\theta<\/script><span>, then <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n544\" cid=\"n544\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 26.478ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.208ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" 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301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-5626-TEX-I-1D462\" d=\"M21 287Q21 295 30 318T55 370T99 420T158 442Q204 442 227 417T250 358Q250 340 216 246T182 105Q182 62 196 45T238 27T291 44T328 78L339 95Q341 99 377 247Q407 367 413 387T427 416Q444 431 463 431Q480 431 488 421T496 402L420 84Q419 79 419 68Q419 43 426 35T447 26Q469 29 482 57T512 145Q514 153 532 153Q551 153 551 144Q550 139 549 130T540 98T523 55T498 17T462 -8Q454 -10 438 -10Q372 -10 347 46Q345 45 336 36T318 21T296 6T267 -6T233 -11Q189 -11 155 7Q103 38 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5626-TEX-I-1D452\" d=\"M39 168Q39 225 58 272T107 350T174 402T244 433T307 442H310Q355 442 388 420T421 355Q421 265 310 237Q261 224 176 223Q139 223 138 221Q138 219 132 186T125 128Q125 81 146 54T209 26T302 45T394 111Q403 121 406 121Q410 121 419 112T429 98T420 82T390 55T344 24T281 -1T205 -11Q126 -11 83 42T39 168ZM373 353Q367 405 305 405Q272 405 244 391T199 357T170 316T154 280T149 261Q149 260 169 260Q282 260 327 284T373 353Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -1401)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1401) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5051.7 -1401 1 2302\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,59)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D449\" xlink:href=\"#MJX-5626-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(769,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-5626-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" 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transform=\"translate(5972.6,0)\"><g data-mml-node=\"mn\" transform=\"translate(1815.4,676)\"><use data-c=\"31\" xlink:href=\"#MJX-5626-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-710)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-5626-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"1D439\" xlink:href=\"#MJX-5626-TEX-I-1D439\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1052.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5626-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1441.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-5626-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(502,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-5626-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(361,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-5626-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(812,0)\"><use data-c=\"1D462\" xlink:href=\"#MJX-5626-TEX-I-1D462\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1384,0)\"><use data-c=\"1D452\" xlink:href=\"#MJX-5626-TEX-I-1D452\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3301.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5626-TEX-N-29\"><\/use><\/g><\/g><rect width=\"3890.8\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1401 1 2302\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,59)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2265<\/mo><mfrac><mn>1<\/mn><mrow><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>\u03b8<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>t<\/mi><mi>r<\/mi><mi>u<\/mi><mi>e<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='properties-of-the-fisher-information'><span>Properties of the Fisher Information:<\/span><\/h4><ul><li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.203ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2299.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-174-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-174-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"1D439\" xlink:href=\"#MJX-174-TEX-I-1D439\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1052.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1441.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-174-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1910.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I_F(\\theta)<\/script><span> depends on given observations <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.225ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 4519.5 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-173-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-173-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-173-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-173-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-173-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-173-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1008.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1453.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-173-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2791.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3236.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-173-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_1,\\dots,x_N<\/script><span> and the unknown parameter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><\/p><\/li><li><p><span>A large value of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.203ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2299.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-174-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-174-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"1D439\" xlink:href=\"#MJX-174-TEX-I-1D439\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1052.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1441.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-174-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1910.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I_F(\\theta)<\/script><span> corresponds to a strong curvature and more information in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.225ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 4519.5 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-173-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-173-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-173-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-173-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-173-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-173-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1008.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1453.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-173-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2791.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3236.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-173-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_1,\\dots,x_N<\/script><span>. A small value of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.203ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2299.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-174-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-174-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"1D439\" xlink:href=\"#MJX-174-TEX-I-1D439\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1052.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1441.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-174-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1910.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I_F(\\theta)<\/script><span> corresponds to a weak curvature and little information in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10.225ex\" height=\"1.439ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 4519.5 636\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-173-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-173-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-173-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-173-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-173-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-173-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1008.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1453.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-173-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2791.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-173-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(3236.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-173-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-173-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">x_1,\\dots,x_N<\/script><span>. <\/span><\/p><\/li><li><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.203ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2299.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-174-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 662Q193 677 195 680T209 683H213Q285 681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z\"><\/path><path id=\"MJX-174-TEX-I-1D439\" d=\"M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z\"><\/path><path id=\"MJX-174-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-174-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-174-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D43C\" xlink:href=\"#MJX-174-TEX-I-1D43C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(473,-150) scale(0.707)\"><use data-c=\"1D439\" xlink:href=\"#MJX-174-TEX-I-1D439\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1052.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-174-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1441.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-174-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1910.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-174-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>I<\/mi><mi>F<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">I_F(\\theta)<\/script><span> is monotonically increasing with the number of independent observation <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msubsup><mi>I<\/mi><mi>F<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>N<\/mi><mo>\u22c5<\/mo><msubsup><mi>I<\/mi><mi>F<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-05-16 23.44.52.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-05-16 23.44.52\"><\/p><h3 id='32-exponential-models'><span>3.2 Exponential Models<\/span><\/h3><p><span>A exponential model is a statistical model with <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n571\" cid=\"n571\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 37.76ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"11.713ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -5.291ex; min-width: 37.76ex;\"><defs><path id=\"MJX-5628-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 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transform=\"translate(462,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-176-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>b<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b_{\\theta}<\/script><span> is for normalization. Then the respective Fisher Information can be directly obtained by <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n573\" cid=\"n573\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 27.901ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.905ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.887ex; min-width: 27.901ex;\"><defs><path id=\"MJX-5629-TEX-I-1D43C\" d=\"M43 1Q26 1 26 10Q26 12 29 24Q34 43 39 45Q42 46 54 46H60Q120 46 136 53Q137 53 138 54Q143 56 149 77T198 273Q210 318 216 344Q286 624 286 626Q284 630 284 631Q274 637 213 637H193Q184 643 189 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transform=\"translate(2136.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-177-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2581.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-177-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3050.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-177-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f_X(x;\\theta) <\/script><span> can be further arranged <\/span><strong><span>such that<\/span><\/strong><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.569ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5113.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-178-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-178-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-178-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-178-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-178-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-178-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-178-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-178-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-178-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-178-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-178-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-178-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1403,0)\"><use data-c=\"28\" xlink:href=\"#MJX-178-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1792,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-178-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2644,0)\"><use data-c=\"29\" xlink:href=\"#MJX-178-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3033,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-178-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3588.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-178-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4644.6,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-178-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>t<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[t(X)]=\\theta<\/script><span>, the unbiased estimator can directly be obtained as <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.114ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4028.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-179-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-179-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-179-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-179-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-179-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-179-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-179-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(981.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-179-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2037.6,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-179-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2398.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-179-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2787.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-179-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3639.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-179-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><mo>=<\/mo><mi>t<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T=t(X)<\/script><span>.<\/span><\/p><h4 id='mean-estimation-example'><span>Mean Estimation Example<\/span><\/h4><p><span>Consider the estimation of the unkown mean value <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.442ex\" height=\"2.084ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 2405.6 921\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.489ex;\"><defs><path id=\"MJX-180-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-180-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-180-TEX-I-1D707\" d=\"M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-180-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(746.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-180-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1802.6,0)\"><use data-c=\"1D707\" xlink:href=\"#MJX-180-TEX-I-1D707\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><mo>=<\/mo><mi>\u03bc<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta=\\mu<\/script><span> of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.57ex\" height=\"2.452ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 5997.8 1083.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-181-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>\u03bc<\/mi><mo>,<\/mo><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X\\sim\\mathcal{N}(\\mu,\\sigma^2)<\/script><span> based on a <\/span><strong><span>single observation<\/span><\/strong><span>.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n577\" cid=\"n577\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" 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xlink:href=\"#MJX-5630-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(6678.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5630-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5630-TEX-N-32\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mover><mo>=<\/mo><mo>!<\/mo><\/mover><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><msqrt><mn>2<\/mn><mi>\u03c0<\/mi><\/msqrt><mi>\u03c3<\/mi><\/mrow><\/mfrac><mi>exp<\/mi><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">[<\/mo><mfrac><mrow><mi>\u03b8<\/mi><mi>x<\/mi><\/mrow><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mfrac><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><msup><mi>\u03b8<\/mi><mn>2<\/mn><\/msup><mrow><mn>2<\/mn><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><mo>+<\/mo><mfrac><mn>1<\/mn><mn>2<\/mn><\/mfrac><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">[<\/mo><mo>\u2212<\/mo><mfrac><mn>1<\/mn><mrow><mn>2<\/mn><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mrow><\/mfrac><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">[<\/mo><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>t<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>b<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u22c5<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mtext>&nbsp;with&nbsp;<\/mtext><mi>t<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>x<\/mi><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>a<\/mi><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u03b8<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msup><mi>\u03c3<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Since <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.374ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 5911.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-182-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 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transform=\"translate(3505.6,0)\"><use data-c=\"1D707\" xlink:href=\"#MJX-182-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4386.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-182-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5442.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-182-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>\u03bc<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E[X]=\\mu=\\theta<\/script><span>, the <\/span><strong><span>single UMVU estimator<\/span><\/strong><span> is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.958ex\" height=\"2.262ex\" role=\"img\" 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data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><msubsup><mi>I<\/mi><mi>F<\/mi><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>A ML estimator is asymptotically efficient if &#8230;<\/span><\/p><\/li><\/ul><h1 id='ii-examples'><span>II. Examples<\/span><\/h1><h2 id='4-ml-principle-for-direction-of-arrival-estimation'><span>4. ML Principle for Direction of Arrival Estimation<\/span><\/h2><h3 id='41-signal-model-and-ml-estimation'><span>4.1 Signal Model and ML Estimation<\/span><\/h3><p><span>We consider the estimation of the Direction Of Arrival(DoA) <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> of an impinging planar waveform by means of an attenna array with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.378ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1051 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-258-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D440\" xlink:href=\"#MJX-258-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>M<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">M<\/script><span> arrays and distance interval <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.176ex\" height=\"1.593ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 520 704\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-186-TEX-I-1D451\" d=\"M366 683Q367 683 438 688T511 694Q523 694 523 686Q523 679 450 384T375 83T374 68Q374 26 402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487H491Q506 153 506 145Q506 140 503 129Q490 79 473 48T445 8T417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157Q33 205 53 255T101 341Q148 398 195 420T280 442Q336 442 364 400Q369 394 369 396Q370 400 396 505T424 616Q424 629 417 632T378 637H357Q351 643 351 645T353 664Q358 683 366 683ZM352 326Q329 405 277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q233 26 290 98L298 109L352 326Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D451\" xlink:href=\"#MJX-186-TEX-I-1D451\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>d<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">d<\/script><span>. The signal vector at time instant <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.817ex\" height=\"1.441ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -626 361 637\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-187-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D461\" xlink:href=\"#MJX-187-TEX-I-1D461\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>t<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">t<\/script><span> is <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n590\" cid=\"n590\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 31.287ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.593ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.731ex; min-width: 31.287ex;\"><defs><path id=\"MJX-5633-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-5633-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5633-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 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0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-73)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtext\"><use data-c=\"1D417\" xlink:href=\"#MJX-5633-TEX-B-1D417\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(869,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5633-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1258,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-5633-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1619,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5633-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2285.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5633-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3341.6,0)\"><use data-c=\"1D709\" xlink:href=\"#MJX-5633-TEX-I-1D709\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(3779.6,0)\"><use data-c=\"1D41A\" xlink:href=\"#MJX-5633-TEX-B-1D41A\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4338.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-5633-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4807.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5633-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5196.6,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-5633-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5557.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5633-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6168.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5633-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(7169,0)\"><use data-c=\"1D40D\" xlink:href=\"#MJX-5633-TEX-B-1D40D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8069,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5633-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8458,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-5633-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8819,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5633-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9485.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-5633-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(10430.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"2102\" xlink:href=\"#MJX-5633-TEX-D-2102\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(755,413) scale(0.707)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5633-TEX-I-1D440\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(11978.7,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5633-TEX-N-A0\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -823 1 1146\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-73)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtext mathvariant=\"bold\">X<\/mtext><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u03be<\/mi><mtext mathvariant=\"bold\">a<\/mtext><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mtext mathvariant=\"bold\">N<\/mtext><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2208<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">C<\/mi><\/mrow><mi>M<\/mi><\/msup><mtext>&nbsp;<\/mtext><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Then the signal at the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.986ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 878 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-262-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-262-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>m<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">m<\/script><span>-th antenna is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n592\" cid=\"n592\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container 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data-mml-node=\"mo\" transform=\"translate(6526.1,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5634-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7304.1,0)\"><use data-c=\"1D457\" xlink:href=\"#MJX-5634-TEX-I-1D457\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(7716.1,0)\"><use data-c=\"32\" xlink:href=\"#MJX-5634-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8216.1,0)\"><use data-c=\"1D70B\" xlink:href=\"#MJX-5634-TEX-I-1D70B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8786.1,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5634-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(9664.1,0)\"><g data-mml-node=\"mi\" transform=\"translate(251.5,676)\"><use data-c=\"1D451\" xlink:href=\"#MJX-5634-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D706\" xlink:href=\"#MJX-5634-TEX-I-1D706\"><\/use><\/g><rect width=\"783\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(10687.1,0)\"><use data-c=\"73\" xlink:href=\"#MJX-5634-TEX-N-73\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-5634-TEX-N-69\" transform=\"translate(394,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5634-TEX-N-6E\" transform=\"translate(672,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11915.1,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-5634-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12081.7,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5634-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12550.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5634-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12939.7,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-5634-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13408.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5634-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13797.7,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-5634-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14158.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5634-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14770,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5634-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(15770.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-5634-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(836,-150) scale(0.707)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5634-TEX-I-1D45A\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(17277,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5634-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17666,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-5634-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18027,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5634-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18416,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5634-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(18860.7,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5634-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(19110.7,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5634-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(20266.5,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5634-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(21322.2,0)\"><use data-c=\"30\" xlink:href=\"#MJX-5634-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(21822.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5634-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(22266.9,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5634-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23605.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5634-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(24050.2,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5634-TEX-I-1D440\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(25323.5,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5634-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(26323.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5634-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1284 1 2068\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-86)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>X<\/mi><mi>m<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>\u03be<\/mi><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>j<\/mi><mn>2<\/mn><mi>\u03c0<\/mi><mi>m<\/mi><mfrac><mi>d<\/mi><mi>\u03bb<\/mi><\/mfrac><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>s<\/mi><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>N<\/mi><mi>m<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>m<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>M<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.319ex\" height=\"1.597ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 583 706\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.027ex;\"><defs><path id=\"MJX-189-TEX-I-1D706\" d=\"M166 673Q166 685 183 694H202Q292 691 316 644Q322 629 373 486T474 207T524 67Q531 47 537 34T546 15T551 6T555 2T556 -2T550 -11H482Q457 3 450 18T399 152L354 277L340 262Q327 246 293 207T236 141Q211 112 174 69Q123 9 111 -1T83 -12Q47 -12 47 20Q47 37 61 52T199 187Q229 216 266 252T321 306L338 322Q338 323 288 462T234 612Q214 657 183 657Q166 657 166 673Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D706\" xlink:href=\"#MJX-189-TEX-I-1D706\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03bb<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\lambda<\/script><span> is the wavelength of the assumed narrowband signal. The signal is assumed to be corrupted by the Gaussian noise vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.158ex\" height=\"2.351ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -789 7583.6 1039\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-190-TEX-B-1D40D\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H171H265Q288 686 297 686T309 684T315 679Q317 676 500 455T684 233V624H576V686H585Q603 683 718 683T851 686H860V624H752V319Q752 15 750 11Q747 4 742 2T718 0H712Q708 0 706 0T700 0T696 1T693 2T690 4T687 7T684 11T679 16T674 23Q671 27 437 311L215 579V62H323V0H314Z\"><\/path><path id=\"MJX-190-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-190-TEX-I-1D461\" d=\"M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z\"><\/path><path id=\"MJX-190-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-190-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-190-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-190-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><path id=\"MJX-190-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-190-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtext\"><use data-c=\"1D40D\" xlink:href=\"#MJX-190-TEX-B-1D40D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(900,0)\"><use data-c=\"28\" xlink:href=\"#MJX-190-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1289,0)\"><use data-c=\"1D461\" xlink:href=\"#MJX-190-TEX-I-1D461\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1650,0)\"><use data-c=\"29\" xlink:href=\"#MJX-190-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2316.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-190-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3372.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-190-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4351.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-190-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4740.6,0)\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-190-TEX-B-1D7CE\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5315.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-190-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5760.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-190-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(748,-150) scale(0.707)\"><use data-c=\"1D40D\" xlink:href=\"#MJX-190-TEX-B-1D40D\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7194.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-190-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtext mathvariant=\"bold\">N<\/mtext><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">0<\/mtext><mo>,<\/mo><msub><mi>C<\/mi><mtext mathvariant=\"bold\">N<\/mtext><\/msub><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\textbf{N}(t)\\sim\\mathcal{N}(\\textbf{0},C_\\textbf{N})<\/script><span>. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.991ex\" height=\"2.057ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -704 438 909\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-191-TEX-I-1D709\" d=\"M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D709\" xlink:href=\"#MJX-191-TEX-I-1D709\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03be<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\xi<\/script><span> represents the attenuation which the transmitted signal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.638ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 1608 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-192-TEX-I-1D460\" d=\"M131 289Q131 321 147 354T203 415T300 442Q362 442 390 415T419 355Q419 323 402 308T364 292Q351 292 340 300T328 326Q328 342 337 354T354 372T367 378Q368 378 368 379Q368 382 361 388T336 399T297 405Q249 405 227 379T204 326Q204 301 223 291T278 274T330 259Q396 230 396 163Q396 135 385 107T352 51T289 7T195 -10Q118 -10 86 19T53 87Q53 126 74 143T118 160Q133 160 146 151T160 120Q160 94 142 76T111 58Q109 57 108 57T107 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stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">s(t)<\/script><span> experiences over the transmission path.<\/span><\/p><p><span>Furthermore, we assume a Uniform Linear Array(ULA) with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.378ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1051 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-258-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>M<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">M<\/script><span> antenna elements, i.e., <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n595\" cid=\"n595\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 44.145ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"13.606ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -6.237ex; min-width: 44.145ex;\"><defs><path id=\"MJX-5635-TEX-B-1D41A\" d=\"M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 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xlink:href=\"#MJX-5635-TEX-N-A0\" transform=\"translate(2195,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8486.1,0)\"><use data-c=\"1D6FC\" xlink:href=\"#MJX-5635-TEX-I-1D6FC\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9403.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5635-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10459.6,0)\"><use data-c=\"65\" xlink:href=\"#MJX-5635-TEX-N-65\"><\/use><use data-c=\"78\" xlink:href=\"#MJX-5635-TEX-N-78\" transform=\"translate(444,0)\"><\/use><use data-c=\"70\" xlink:href=\"#MJX-5635-TEX-N-70\" transform=\"translate(972,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11987.6,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-5635-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11987.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5635-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12376.6,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5635-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(13154.6,0)\"><use data-c=\"1D457\" xlink:href=\"#MJX-5635-TEX-I-1D457\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(13566.6,0)\"><use data-c=\"32\" xlink:href=\"#MJX-5635-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(14066.6,0)\"><use data-c=\"1D70B\" xlink:href=\"#MJX-5635-TEX-I-1D70B\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(14636.6,0)\"><g data-mml-node=\"mi\" transform=\"translate(251.5,676)\"><use data-c=\"1D451\" xlink:href=\"#MJX-5635-TEX-I-1D451\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D706\" xlink:href=\"#MJX-5635-TEX-I-1D706\"><\/use><\/g><rect width=\"783\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"mi\" transform=\"translate(15659.6,0)\"><use data-c=\"73\" xlink:href=\"#MJX-5635-TEX-N-73\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-5635-TEX-N-69\" transform=\"translate(394,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5635-TEX-N-6E\" transform=\"translate(672,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16887.6,0)\"><use data-c=\"2061\" xlink:href=\"#MJX-5635-TEX-N-2061\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17054.3,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5635-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17523.3,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5635-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -3256.9 1 6013.8\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mtext mathvariant=\"bold\">a<\/mtext><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msup><mi>\u03b1<\/mi><mn>0<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><msup><mi>\u03b1<\/mi><mn>1<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mo>\u22ee<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd><msup><mi>\u03b1<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>,<\/mo><mtext>&nbsp;with&nbsp;<\/mtext><mi>\u03b1<\/mi><mo>=<\/mo><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mi>j<\/mi><mn>2<\/mn><mi>\u03c0<\/mi><mfrac><mi>d<\/mi><mi>\u03bb<\/mi><\/mfrac><mi>sin<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>In the case of a single observation and AWGN <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"17.799ex\" height=\"2.565ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 7867.1 1133.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.679ex;\"><defs><path id=\"MJX-194-TEX-B-1D40D\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H171H265Q288 686 297 686T309 684T315 679Q317 676 500 455T684 233V624H576V686H585Q603 683 718 683T851 686H860V624H752V319Q752 15 750 11Q747 4 742 2T718 0H712Q708 0 706 0T700 0T696 1T693 2T690 4T687 7T684 11T679 16T674 23Q671 27 437 311L215 579V62H323V0H314Z\"><\/path><path id=\"MJX-194-TEX-N-28\" d=\"M94 250Q94 319 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178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-194-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-194-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 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data-mml-node=\"mo\" transform=\"translate(2316.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-194-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3372.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-194-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4351.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-194-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(4740.6,0)\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-194-TEX-B-1D7CE\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5315.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-194-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(5760.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-194-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-194-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-300) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-194-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(7042.1,0)\"><use data-c=\"1D408\" xlink:href=\"#MJX-194-TEX-B-1D408\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7478.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-194-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mtext mathvariant=\"bold\">N<\/mtext><mo stretchy=\"false\">(<\/mo><mi>t<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">0<\/mtext><mo>,<\/mo><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><mtext mathvariant=\"bold\">I<\/mtext><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\textbf{N}(t)\\sim\\mathcal{N}(\\textbf{0},\\sigma^2_N\\textbf{I})<\/script><span>, we have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n597\" cid=\"n597\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"51.174ex\" height=\"20.522ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -4785.4 22618.8 9070.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -9.695ex;\"><defs><path id=\"MJX-5636-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><path id=\"MJX-5636-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5636-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-5636-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-5636-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-5636-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 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358H408L405 341Q393 269 393 222Q393 170 402 129T421 65T431 37Q431 20 417 5T381 -10Q370 -10 363 -7T347 17T331 77Q330 86 330 121Q330 170 339 226T357 318T367 358H269L268 354Q268 351 249 275T206 114T175 17Q164 -11 132 -11Z\"><\/path><path id=\"MJX-5636-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-5636-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 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683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-5636-TEX-N-65\" d=\"M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z\"><\/path><path id=\"MJX-5636-TEX-N-78\" d=\"M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z\"><\/path><path id=\"MJX-5636-TEX-N-70\" d=\"M36 -148H50Q89 -148 97 -134V-126Q97 -119 97 -107T97 -77T98 -38T98 6T98 55T98 106Q98 140 98 177T98 243T98 296T97 335T97 351Q94 370 83 376T38 385H20V408Q20 431 22 431L32 432Q42 433 61 434T98 436Q115 437 135 438T165 441T176 442H179V416L180 390L188 397Q247 441 326 441Q407 441 464 377T522 216Q522 115 457 52T310 -11Q242 -11 190 33L182 40V-45V-101Q182 -128 184 -134T195 -145Q216 -148 244 -148H260V-194H252L228 -193Q205 -192 178 -192T140 -191Q37 -191 28 -194H20V-148H36ZM424 218Q424 292 390 347T305 402Q234 402 182 337V98Q222 26 294 26Q345 26 384 80T424 218Z\"><\/path><path id=\"MJX-5636-TEX-N-2061\" d=\"\"><\/path><path id=\"MJX-5636-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5636-TEX-I-1D709\" d=\"M268 632Q268 704 296 704Q314 704 314 687Q314 682 311 664T308 635T309 620V616H315Q342 619 360 619Q443 619 443 586Q439 548 358 546H344Q326 546 317 549T290 566Q257 550 226 505T195 405Q195 381 201 364T211 342T218 337Q266 347 298 347Q375 347 375 314Q374 297 359 288T327 277T280 275Q234 275 208 283L195 286Q149 260 119 214T88 130Q88 116 90 108Q101 79 129 63T229 20Q238 17 243 15Q337 -21 354 -33Q383 -53 383 -94Q383 -137 351 -171T273 -205Q240 -205 202 -190T158 -167Q156 -163 156 -159Q156 -151 161 -146T176 -140Q182 -140 189 -143Q232 -168 274 -168Q286 -168 292 -165Q313 -151 313 -129Q313 -112 301 -104T232 -75Q214 -68 204 -64Q198 -62 171 -52T136 -38T107 -24T78 -8T56 12T36 37T26 66T21 103Q21 149 55 206T145 301L154 307L148 313Q141 319 136 323T124 338T111 358T103 382T99 413Q99 471 143 524T259 602L271 607Q268 618 268 632Z\"><\/path><path id=\"MJX-5636-TEX-B-1D41A\" d=\"M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 196V77Q493 63 529 62Q549 62 553 57T558 31Q558 9 552 5T514 0H497H481Q375 0 367 56L356 46Q300 -6 210 -6Q130 -6 81 30T32 121Q32 188 111 226T332 272H350V292Q350 313 348 327T337 361T306 391T248 402T194 399H189Q204 376 204 354Q204 327 187 306T134 284Q97 284 81 305T64 349ZM164 121Q164 89 186 67T238 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stretchy=\"false\">)<\/mo><mi>H<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">x<\/mtext><mo>\u2212<\/mo><mi>\u03be<\/mi><mtext mathvariant=\"bold\">a<\/mtext><mi>s<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><\/mfrac><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/munder><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">x<\/mtext><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/munder><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">x<\/mtext><mo>\u2212<\/mo><mi>\u03be<\/mi><mtext mathvariant=\"bold\">a<\/mtext><mi>s<\/mi><msup><mo stretchy=\"false\">)<\/mo><mi>H<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">x<\/mtext><mo>\u2212<\/mo><mi>\u03be<\/mi><mtext mathvariant=\"bold\">a<\/mtext><mi>s<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/munder><msup><mtext mathvariant=\"bold\">x<\/mtext><mi>H<\/mi><\/msup><mtext mathvariant=\"bold\">x<\/mtext><mo>\u2212<\/mo><mn>2<\/mn><mi>R<\/mi><mi>e<\/mi><mo fence=\"false\" stretchy=\"false\">{<\/mo><mi>\u03be<\/mi><msup><mtext mathvariant=\"bold\">x<\/mtext><mi>H<\/mi><\/msup><mtext mathvariant=\"bold\">a<\/mtext><mi>s<\/mi><mo fence=\"false\" stretchy=\"false\">}<\/mo><mo>+<\/mo><mi>\u03be<\/mi><msup><mtext mathvariant=\"bold\">a<\/mtext><mi>H<\/mi><\/msup><mtext mathvariant=\"bold\">a<\/mtext><msup><mi>s<\/mi><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>=<\/mo><mo>!<\/mo><\/mover><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/munder><mi>R<\/mi><mi>e<\/mi><mo fence=\"false\" stretchy=\"false\">{<\/mo><msup><mtext mathvariant=\"bold\">x<\/mtext><mi>H<\/mi><\/msup><mtext mathvariant=\"bold\">a<\/mtext><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><strong><span>!<\/span><\/strong><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"24.981ex\" height=\"2.099ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 11041.6 928\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-195-TEX-I-1D6FC\" d=\"M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 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data-mml-node=\"mtext\" transform=\"translate(6375.6,0)\"><use data-c=\"1D41A\" xlink:href=\"#MJX-195-TEX-B-1D41A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7212.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-195-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8268.1,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-195-TEX-I-1D440\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9541.4,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-195-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(10541.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-195-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mi>\u03b1<\/mi><mo>\u2217<\/mo><\/msup><mi>\u03b1<\/mi><mo>=<\/mo><mn>1<\/mn><mo stretchy=\"false\">\u21d2<\/mo><msup><mtext mathvariant=\"bold\">a<\/mtext><mi>H<\/mi><\/msup><mtext mathvariant=\"bold\">a<\/mtext><mo>=<\/mo><mi>M<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\alpha^*\\alpha=1\\Rightarrow \\textbf{a}^H\\textbf{a}=M-1<\/script><span>.<\/span><\/p><h3 id='42-cramer-rao-bound-for-doa-estimation'><span>4.2 Cramer-Rao Bound for DOA Estimation<\/span><\/h3><p><span>The likelihood function of the given estimation problem obviously belongs to the family of exponential distributions, where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is the parameter to be estimated.<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n601\" cid=\"n601\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" 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stretchy=\"false\">)<\/mo><mi>H<\/mi><\/msup><mfrac><mrow><mi>\u2202<\/mi><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mtext mathvariant=\"bold\">t<\/mtext><mo stretchy=\"false\">(<\/mo><mtext mathvariant=\"bold\">X<\/mtext><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><\/mrow><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mn>2<\/mn><mfrac><mrow><msup><mi>\u03be<\/mi><mn>2<\/mn><\/msup><msup><mi>s<\/mi><mn>2<\/mn><\/msup><\/mrow><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><\/mfrac><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mfrac><mrow><mi>\u2202<\/mi><mtext mathvariant=\"bold\">a<\/mtext><\/mrow><mrow><mi>\u2202<\/mi><mi>\u03b8<\/mi><\/mrow><\/mfrac><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mn>2<\/mn><\/msup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo data-mjx-alternate=\"1\">\u221d<\/mo><mo stretchy=\"false\">(<\/mo><mfrac><mi>d<\/mi><mi>\u03bb<\/mi><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mfrac><mrow><msup><mi>\u03be<\/mi><mn>2<\/mn><\/msup><msup><mi>s<\/mi><mn>2<\/mn><\/msup><\/mrow><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>cos<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><msup><mi>M<\/mi><mn>3<\/mn><\/msup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h1 id='iii-estimation-of-random-variables'><span>III. Estimation of Random Variables<\/span><\/h1><h2 id='5-bayesian-estimation'><span>5. Bayesian Estimation<\/span><\/h2><p><span>In the previous ML principle, we don&#39;t have any statistical information about the parameter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>, and only use the likelihood function to decide <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"2.45ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1073 469 1083\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-198-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-198-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-198-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(317.8,279) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-198-TEX-N-5E\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat{\\theta}<\/script><span>.<\/span><\/p><p><span>The Bayesian Estimation Method is based on a specific Statistical Model for the unknown parameter, here, it is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>. So we assume the statistical information about <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span>, consequently, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is now considered as a random variable, i.e.,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n606\" cid=\"n606\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 17.221ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 17.221ex;\"><defs><path id=\"MJX-5638-TEX-N-50\" d=\"M130 622Q123 629 119 631T103 634T60 637H27V683H214Q237 683 276 683T331 684Q419 684 471 671T567 616Q624 563 624 489Q624 421 573 372T451 307Q429 302 328 301H234V181Q234 62 237 58Q245 47 304 46H337V0H326Q305 3 182 3Q47 3 38 0H27V46H60Q102 47 111 49T130 61V622ZM507 488Q507 514 506 528T500 564T483 597T450 620T397 635Q385 637 307 637H286Q237 637 234 628Q231 624 231 483V342H302H339Q390 342 423 349T481 382Q507 411 507 488Z\"><\/path><path id=\"MJX-5638-TEX-N-44\" d=\"M130 622Q123 629 119 631T103 634T60 637H27V683H228Q399 682 419 682T461 676Q504 667 546 641T626 573T685 470T708 336Q708 210 634 116T442 3Q429 1 228 0H27V46H60Q102 47 111 49T130 61V622ZM593 338Q593 439 571 501T493 602Q439 637 355 637H322H294Q238 637 234 628Q231 624 231 344Q231 62 232 59Q233 49 248 48T339 46H350Q456 46 515 95Q561 133 577 191T593 338Z\"><\/path><path id=\"MJX-5638-TEX-N-46\" d=\"M128 619Q121 626 117 628T101 631T58 634H25V680H582V676Q584 670 596 560T610 444V440H570V444Q563 493 561 501Q555 538 543 563T516 601T477 622T431 631T374 633H334H286Q252 633 244 631T233 621Q232 619 232 490V363H284Q287 363 303 363T327 364T349 367T372 373T389 385Q407 403 410 459V480H450V200H410V221Q407 276 389 296Q381 303 371 307T348 313T327 316T303 317T284 317H232V189L233 61Q240 54 245 52T270 48T333 46H360V0H348Q324 3 182 3Q51 3 36 0H25V46H58Q100 47 109 49T128 61V619Z\"><\/path><path id=\"MJX-5638-TEX-N-3A\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60Z\"><\/path><path id=\"MJX-5638-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-5638-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-5638-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-5638-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5638-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-5638-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-5638-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-5638-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3005.9 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtext\"><use data-c=\"50\" xlink:href=\"#MJX-5638-TEX-N-50\"><\/use><use data-c=\"44\" xlink:href=\"#MJX-5638-TEX-N-44\" transform=\"translate(681,0)\"><\/use><use data-c=\"46\" xlink:href=\"#MJX-5638-TEX-N-46\" transform=\"translate(1445,0)\"><\/use><use data-c=\"3A\" xlink:href=\"#MJX-5638-TEX-N-3A\" transform=\"translate(2098,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-5638-TEX-N-A0\" transform=\"translate(2376,0)\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2626,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-5638-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(523,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-5638-TEX-N-398\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3749.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5638-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4138.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5638-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4607.1,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-5638-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5051.8,0)\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5638-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5622.8,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5638-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtext>PDF:&nbsp;<\/mtext><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo>;<\/mo><mi>\u03c3<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.292ex\" height=\"1ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -431 571 442\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-202-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-202-TEX-I-1D70E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03c3<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma<\/script><span> is the parameter for the statistical model of R.V. <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 778 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-224-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-224-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Theta<\/script><span>, for example, a specific PDF.<\/span><\/p><p><span>Furthermore, instead of using notation <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.781ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3439.1 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-204-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-204-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-204-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-204-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-204-TEX-N-3B\" d=\"M78 370Q78 394 95 412T138 430Q162 430 180 414T199 371Q199 346 182 328T139 310T96 327T78 370ZM78 60Q78 85 94 103T137 121Q202 121 202 8Q202 -44 183 -94T144 -169T118 -194Q115 -194 106 -186T95 -174Q94 -171 107 -155T137 -107T160 -38Q161 -32 162 -22T165 -4T165 4Q165 5 161 4T142 0Q110 0 94 18T78 60Z\"><\/path><path id=\"MJX-204-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-204-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-204-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-204-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1175.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-204-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1564.5,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-204-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2136.5,0)\"><use data-c=\"3B\" xlink:href=\"#MJX-204-TEX-N-3B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2581.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-204-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3050.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-204-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f_X(x;\\theta)<\/script><span>, we have to use the conditional PDF, because <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.659ex\" height=\"2.059ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 1175.5 910\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-205-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z\"><\/path><path id=\"MJX-205-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-205-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-205-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>f<\/mi><mi>X<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f_X<\/script><span> can only be obtained after <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 778 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-224-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-224-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Theta<\/script><span> is realized and <\/span><strong><span>a certain value of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is used<\/span><\/strong><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n609\" cid=\"n609\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 30.159ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.495ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.682ex; min-width: 30.159ex;\"><defs><path id=\"MJX-5639-TEX-N-63\" d=\"M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 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340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-5639-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5639-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 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-250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -801.4)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 801.4) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5865.1 -801.4 1 1102.8\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,51.4)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtext\"><use data-c=\"63\" xlink:href=\"#MJX-5639-TEX-N-63\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5639-TEX-N-6F\" transform=\"translate(444,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5639-TEX-N-6E\" transform=\"translate(944,0)\"><\/use><use data-c=\"64\" xlink:href=\"#MJX-5639-TEX-N-64\" transform=\"translate(1500,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-5639-TEX-N-69\" transform=\"translate(2056,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-5639-TEX-N-74\" transform=\"translate(2334,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-5639-TEX-N-69\" transform=\"translate(2723,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5639-TEX-N-6F\" transform=\"translate(3001,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5639-TEX-N-6E\" transform=\"translate(3501,0)\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-5639-TEX-N-61\" transform=\"translate(4057,0)\"><\/use><use data-c=\"6C\" xlink:href=\"#MJX-5639-TEX-N-6C\" transform=\"translate(4557,0)\"><\/use><use data-c=\"20\" xlink:href=\"#MJX-5639-TEX-N-20\" transform=\"translate(4835,0)\"><\/use><use data-c=\"50\" xlink:href=\"#MJX-5639-TEX-N-50\" transform=\"translate(5085,0)\"><\/use><use data-c=\"44\" xlink:href=\"#MJX-5639-TEX-N-44\" transform=\"translate(5766,0)\"><\/use><use data-c=\"46\" xlink:href=\"#MJX-5639-TEX-N-46\" transform=\"translate(6530,0)\"><\/use><use data-c=\"3A\" xlink:href=\"#MJX-5639-TEX-N-3A\" transform=\"translate(7183,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-5639-TEX-N-A0\" transform=\"translate(7461,0)\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7711,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-5639-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(523,-176.4) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5639-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5639-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5639-TEX-N-398\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(9633.2,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5639-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10022.2,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-5639-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10594.2,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5639-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(10872.2,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-5639-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11341.2,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5639-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -801.4 1 1102.8\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,51.4)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtext>conditional PDF:&nbsp;<\/mtext><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Now, the MSE of estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.647ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2054 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-208-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-208-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-208-TEX-I-1D465\" d=\"M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-208-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-208-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(704,0)\"><use data-c=\"28\" xlink:href=\"#MJX-208-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1093,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-208-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1665,0)\"><use data-c=\"29\" xlink:href=\"#MJX-208-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T(x)<\/script><span> is calculated with respect to both random variables <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4206-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 778 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-224-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-224-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Theta<\/script><span>,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n611\" cid=\"n611\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 76.292ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.142ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.006ex; min-width: 76.292ex;\"><defs><path id=\"MJX-5640-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-5640-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-5640-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 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data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo>:<\/mo><mi>x<\/mi><mo stretchy=\"false\">\u21a6<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo>=<\/mo><mi>x<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mi>\u03b8<\/mi><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>\u03b8<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><strong><span>Theorem<\/span><\/strong><\/p><\/li><\/ul><p><span>The conditional mean estimator (Bayes estimator) <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.406ex\" height=\"1.906ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1947.6 842.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-243-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 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638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-243-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-243-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-243-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_{CM}<\/script><span> is <\/span><strong><span>MSE optimal<\/span><\/strong><span>, i.e., minimizes the MSE cost criterion:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" 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transform=\"translate(14186.6,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5642-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15186.8,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5642-TEX-N-398\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(15964.8,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-5642-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(422,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5642-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(16790.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5642-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(17068.4,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5642-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17846.4,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5642-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(18124.4,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5642-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -817 1 1133.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-67)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi>\u03b5<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><msub><mi>E<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><mi>T<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>Alternative cost criterion<\/span><\/p><\/li><\/ul><p><span>Although the mean MSE is the most popular cost criterion, other criteria have been proposed and applied. 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stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><h3 id='52-binomial-experiment'><span>5.2 Binomial Experiment<\/span><\/h3><p><span>The ML estimator is obtained as<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n627\" cid=\"n627\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 55.571ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"4.081ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.475ex; min-width: 55.571ex;\"><defs><path 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data-mml-node=\"mfrac\" transform=\"translate(21634.4,0)\"><g data-mml-node=\"mi\" transform=\"translate(378,676)\"><use data-c=\"1D465\" xlink:href=\"#MJX-5645-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5645-TEX-I-1D441\"><\/use><\/g><rect width=\"1088\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1152 1 1804\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,34)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>:<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">\u21a6<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>=<\/mo><msub><mi>arg<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>\u03b8<\/mi><\/mrow><\/msub><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">max<\/mo><mi>log<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mi>L<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>;<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mfrac><mi>x<\/mi><mi>N<\/mi><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Now we assume <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.511ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 4203.7 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-213-TEX-I-1D453\" d=\"M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 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663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-213-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(523,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-213-TEX-N-398\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1123.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-213-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1512.1,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-213-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1981.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-213-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2647.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-213-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3703.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-213-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f_{\\Theta}(\\theta)=1<\/script><span> if <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.353ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3692.2 1000\" 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id=\"MJX-214-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-214-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(746.8,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-214-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1691.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-214-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1969.6,0)\"><use data-c=\"30\" xlink:href=\"#MJX-214-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2469.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-214-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2914.2,0)\"><use data-c=\"31\" xlink:href=\"#MJX-214-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3414.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-214-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta\\in[0,1]<\/script><span>, then the MSE estimator is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n629\" cid=\"n629\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"32.914ex\" height=\"14.918ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -3546.9 14548 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display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>N<\/mi><mi>x<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><msup><mi>\u03b8<\/mi><mi>x<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>\u03b8<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><msub><mi>p<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>\u03b8<\/mi><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>\u03b8<\/mi><mo>\u2208<\/mo><mo stretchy=\"false\">[<\/mo><mn>0<\/mn><mo>,<\/mo><mn>1<\/mn><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>N<\/mi><mi>x<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><msup><mi>\u03b8<\/mi><mi>x<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><msub><mi>f<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>\u03b8<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mo data-mjx-texclass=\"OP\">\u222b<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"OPEN\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">(<\/mo><\/mrow><mfrac linethickness=\"0\"><mi>N<\/mi><mi>x<\/mi><\/mfrac><mrow data-mjx-texclass=\"CLOSE\"><mo minsize=\"2.047em\" maxsize=\"2.047em\">)<\/mo><\/mrow><\/mrow><\/mrow><msup><mi>\u03b8<\/mi><mi>x<\/mi><\/msup><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>\u03b8<\/mi><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/msup><mi>d<\/mi><mi>\u03b8<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>We get <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n633\" cid=\"n633\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" 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641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-5648-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 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404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z\"><\/path><path id=\"MJX-5648-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5648-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path 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mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><mo stretchy=\"false\">)<\/mo><mi>d<\/mi><mi>\u03b8<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mrow><msub><mi>p<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mfrac><mrow><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><mfrac><mn>1<\/mn><mrow><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mi>N<\/mi><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mfrac><mn>1<\/mn><mrow><mi>N<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-05-17 22.37.06.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-05-17 22.37.06\"><\/p><h3 id='53-mean-estimation-example'><span>5.3 Mean Estimation Example<\/span><\/h3><p><span>We consider the estimation of the unknown mean value <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-228-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-228-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\u03b8<\/script><span> of Gaussian Distributed random variable<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n637\" 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xlink:href=\"#MJX-5649-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -928.7 1 1357.3\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,44.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>X<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo>,<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi 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xlink:href=\"#MJX-216-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2111.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-216-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3090.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-216-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3479.6,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-216-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4357.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-216-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(4802.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-216-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-216-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-315.5) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-216-TEX-N-398\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6006.4,0)\"><use data-c=\"29\" xlink:href=\"#MJX-216-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mi>m<\/mi><mo>,<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Theta\\sim\\mathcal{N}(m,\\sigma^2_{\\Theta})<\/script><span>, given <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use 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data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-218-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-218-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_1,\\dots,X_N<\/script><span> taken from the joint conditional likelihood of iid observations<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n639\" cid=\"n639\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" 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transform=\"translate(10387,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5650-TEX-N-6F\" transform=\"translate(10665,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5650-TEX-N-6E\" transform=\"translate(11165,0)\"><\/use><use data-c=\"3A\" xlink:href=\"#MJX-5650-TEX-N-3A\" transform=\"translate(11721,0)\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-1040.8)\"><g data-mml-node=\"mtd\"><\/g><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5650-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1129.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2185.6,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3185.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5650-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4186,0)\"><use data-c=\"1D441\" 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data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(11595.6,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5650-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-4320.5)\"><g data-mml-node=\"mtd\"><\/g><g data-mml-node=\"mtd\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5650-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5650-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-250) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5650-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mover\" transform=\"translate(1534.2,0)\"><g data-mml-node=\"mo\" transform=\"translate(1390.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(0,831.2) scale(0.707)\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(778,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1056,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5650-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(1944,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5650-TEX-N-A0\"><\/use><use data-c=\"75\" xlink:href=\"#MJX-5650-TEX-N-75\" transform=\"translate(250,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5650-TEX-N-6E\" transform=\"translate(806,0)\"><\/use><use data-c=\"63\" xlink:href=\"#MJX-5650-TEX-N-63\" transform=\"translate(1362,0)\"><\/use><use data-c=\"6F\" xlink:href=\"#MJX-5650-TEX-N-6F\" transform=\"translate(1806,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-5650-TEX-N-72\" transform=\"translate(2306,0)\"><\/use><use data-c=\"72\" xlink:href=\"#MJX-5650-TEX-N-72\" transform=\"translate(2698,0)\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(5371.6,0)\"><use data-c=\"1D449\" xlink:href=\"#MJX-5650-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6140.6,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-5650-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6669.6,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-5650-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7120.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5650-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7398.6,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8176.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5650-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8676.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5650-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9677,0)\"><use 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transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5650-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-265.5) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15629.9,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5650-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(16630.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5650-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5650-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-250) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5650-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-5785.5)\"><g data-mml-node=\"mtd\"><\/g><g 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data-mml-node=\"mo\" transform=\"translate(4638.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5694.2,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-5650-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6454.2,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-5650-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(6939.2,0)\"><use data-c=\"1D463\" xlink:href=\"#MJX-5650-TEX-I-1D463\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7424.2,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5650-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7702.2,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8480.2,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8924.9,0)\"><use data-c=\"398\" 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transform=\"translate(15154.9,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5650-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15432.9,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16210.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16655.6,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17433.6,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5650-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(17933.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5650-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(18934,0)\"><use data-c=\"1D436\" xlink:href=\"#MJX-5650-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(19694,0)\"><use data-c=\"1D45C\" xlink:href=\"#MJX-5650-TEX-I-1D45C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20179,0)\"><use data-c=\"1D463\" xlink:href=\"#MJX-5650-TEX-I-1D463\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(20664,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5650-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(20942,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(21720,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5650-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(22164.7,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5650-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23052.7,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5650-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(23608.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(24664.2,0)\"><use data-c=\"1D449\" xlink:href=\"#MJX-5650-TEX-I-1D449\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(25433.2,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-5650-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(25962.2,0)\"><use data-c=\"1D45F\" xlink:href=\"#MJX-5650-TEX-I-1D45F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(26413.2,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5650-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(26691.2,0)\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(27469.2,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5650-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(28025,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5650-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(29080.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5650-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5650-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-265.5) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-5650-TEX-N-398\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><mtext>Conditional Distribution:<\/mtext><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi>X<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><mo>+<\/mo><mi>N<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mi>\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><\/mrow><\/msub><mo>=<\/mo><mi>\u03b8<\/mi><mo>,<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><\/mtd><\/mtr><mtr><mtd><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mtext>Unconditional Distribution:<\/mtext><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi>X<\/mi><mo>=<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>+<\/mo><mi>N<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>+<\/mo><mi>N<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>m<\/mi><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msubsup><mi>\u03c3<\/mi><mi>X<\/mi><mn>2<\/mn><\/msubsup><mover><mo>=<\/mo><mrow><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>N<\/mi><mtext>&nbsp;uncorr<\/mtext><\/mrow><\/mover><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi>N<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi>C<\/mi><mi>o<\/mi><mi>v<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>C<\/mi><mi>o<\/mi><mi>v<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>+<\/mo><mi>N<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>C<\/mi><mi>o<\/mi><mi>v<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mi>C<\/mi><mi>o<\/mi><mi>v<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>N<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='conditional-mean-estimator-two-steps-needed'><span>Conditional mean estimator (two steps needed):<\/span><\/h4><ul><li><p><span>Computing conditional PDF<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" 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mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><mi>\u03b8<\/mi><mo stretchy=\"false\">(<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>+<\/mo><mi>m<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>\u221d<\/mo><mi>exp<\/mi><mo data-mjx-texclass=\"NONE\">\u2061<\/mo><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mfrac><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><mrow><mn>2<\/mn><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>\u03b8<\/mi><mo>\u2212<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>+<\/mo><mi>m<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><msup><mo stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mtext>directly read out:<\/mtext><\/mtd><mtd><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msub><mi>\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>x<\/mi><mi>i<\/mi><\/msub><mo>+<\/mo><mi>m<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><mrow><mo stretchy=\"false\">(<\/mo><mi>N<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mn>5.34<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/mrow><mn>2<\/mn><\/msubsup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi>N<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>Computing conditional mean<\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n648\" cid=\"n648\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"53.08ex\" height=\"18.182ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -4268.3 23461.4 8036.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -8.525ex;\"><defs><path id=\"MJX-5652-TEX-I-1D703\" 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data-mml-node=\"mfrac\" transform=\"translate(1333.6,0)\"><g data-mml-node=\"mn\" transform=\"translate(414,676)\"><use data-c=\"31\" xlink:href=\"#MJX-5652-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(220,-686)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5652-TEX-I-1D441\"><\/use><\/g><rect width=\"1088\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><g data-mml-node=\"munderover\" transform=\"translate(2828.2,0)\"><g data-mml-node=\"mo\"><use data-c=\"2211\" xlink:href=\"#MJX-5652-TEX-LO-2211\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(148.2,-1087.9) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D456\" xlink:href=\"#MJX-5652-TEX-I-1D456\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(345,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5652-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1123,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5652-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(408,1150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5652-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(4438.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D465\" xlink:href=\"#MJX-5652-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(605,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5652-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msub><mi>x<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>x<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>N<\/mi><\/mrow><\/mfrac><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>+<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>N<\/mi><\/mrow><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>N<\/mi><\/mrow><\/mfrac><mi>m<\/mi><\/mtd><\/mtr><mtr><mtd><mtext>with&nbsp;<\/mtext><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mn>1<\/mn><mi>N<\/mi><\/mfrac><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>x<\/mi><mi>i<\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='discussion'><span>Discussion<\/span><\/h4><ul><li><p><span>Given large <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> or small conditional variance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.527ex\" height=\"3.071ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 2884.9 1357.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.184ex;\"><defs><path id=\"MJX-223-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-223-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-223-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-223-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-223-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-223-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-223-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-223-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-223-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-347) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-223-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-223-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"398\" xlink:href=\"#MJX-223-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1908,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-223-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2686,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-223-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma^2_{X|\\Theta=\\theta}<\/script><span> or large variance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"2.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1204.1 1165\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.749ex;\"><defs><path id=\"MJX-222-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-222-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-222-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-222-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-222-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-315.5) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-222-TEX-N-398\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma^2_{\\Theta}<\/script><span>, it is recommended to rely on the ML estimator.<\/span><\/p><\/li><li><p><span>Given small variance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.724ex\" height=\"2.636ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 1204.1 1165\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.749ex;\"><defs><path id=\"MJX-222-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-222-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-222-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-222-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-222-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-315.5) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-222-TEX-N-398\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma^2_{\\Theta}<\/script><span> or large conditional variance <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.527ex\" height=\"3.071ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 2884.9 1357.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.184ex;\"><defs><path id=\"MJX-223-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-223-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-223-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-223-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-223-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-223-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-223-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-223-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-223-TEX-N-32\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(604,-347) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-223-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(852,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-223-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1130,0)\"><use data-c=\"398\" xlink:href=\"#MJX-223-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1908,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-223-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2686,0)\"><use data-c=\"1D703\" xlink:href=\"#MJX-223-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\sigma^2_{X|\\Theta=\\theta}<\/script><span>, it is recommended to rely on the mean value of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.76ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 778 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-224-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-224-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\Theta<\/script><span>.<\/span><\/p><\/li><\/ul><h4 id='minimum-mean-square-error'><span>Minimum Mean Square Error<\/span><\/h4><p><span>MSE is minimized at <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.406ex\" height=\"1.906ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1947.6 842.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-243-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-243-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-243-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-243-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-243-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-243-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_{CM}<\/script><span>, and is given by<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n657\" cid=\"n657\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"79.873ex\" height=\"27.015ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -6220.4 35303.9 11940.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -12.942ex;\"><defs><path id=\"MJX-5653-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-5653-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-5653-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-5653-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5653-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5653-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 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-22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-5653-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 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mathvariant=\"normal\">\u0398<\/mi><mo>+<\/mo><msup><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">)<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><msubsup><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mn>2<\/mn><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><msup><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><msup><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><msubsup><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mi>X<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>E<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><\/msub><mo stretchy=\"false\">[<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi>N<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>which can be obtained once we have 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stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>V<\/mi><mi>a<\/mi><mi>r<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='54-jointly-gaussian-random-variables---multivariate-case'><span>5.4 Jointly Gaussian Random Variables &#8211; Multivariate Case<\/span><\/h3><p><span>Given random vectors and the covariance matrix from the joint distribution:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n662\" cid=\"n662\" mdtype=\"math_block\" data-math-tag-before=\"0\" data-math-tag-after=\"0\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" 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data-mml-node=\"mtext\"><use data-c=\"1D402\" xlink:href=\"#MJX-5655-TEX-B-1D402\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6AF\" xlink:href=\"#MJX-5655-TEX-B-1D6AF\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(894,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5655-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(1172,0)\"><use data-c=\"1D417\" xlink:href=\"#MJX-5655-TEX-B-1D417\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(3762.7,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mtext\"><use data-c=\"1D402\" xlink:href=\"#MJX-5655-TEX-B-1D402\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6AF\" xlink:href=\"#MJX-5655-TEX-B-1D6AF\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6242.4,0) translate(0 -0.5)\"><use data-c=\"5D\" xlink:href=\"#MJX-5655-TEX-S3-5D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">N<\/mi><\/mrow><\/mtd><\/mtr><mtr><mtd><mtext mathvariant=\"bold\">X<\/mtext><\/mtd><mtd><mi><\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mtext mathvariant=\"bold\">X<\/mtext><\/mrow><\/msub><mo>,<\/mo><msub><mtext mathvariant=\"bold\">C<\/mtext><mtext mathvariant=\"bold\">X<\/mtext><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mi mathvariant=\"bold\">\u0398<\/mi><\/mtd><mtd><mi><\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><\/msub><mo>,<\/mo><msub><mtext mathvariant=\"bold\">C<\/mtext><mi mathvariant=\"bold\">\u0398<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi mathvariant=\"bold-italic\">C<\/mi><mi mathvariant=\"bold-italic\">Z<\/mi><\/msub><\/mtd><mtd><mi><\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mtext mathvariant=\"bold\">C<\/mtext><mtext mathvariant=\"bold\">X<\/mtext><\/msub><\/mtd><mtd><msub><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mtext mathvariant=\"bold\">X<\/mtext><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><mo>,<\/mo><mtext mathvariant=\"bold\">X<\/mtext><\/mrow><\/msub><\/mtd><mtd><msub><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>the multivariate conditional mean estimator is obtained as:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n664\" cid=\"n664\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"57.402ex\" height=\"2.701ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -847 25371.7 1194\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.785ex;\"><defs><path id=\"MJX-5657-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 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xlink:href=\"#MJX-5657-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(12959.5,0)\"><use data-c=\"35\" xlink:href=\"#MJX-5657-TEX-N-35\"><\/use><use data-c=\"2E\" xlink:href=\"#MJX-5657-TEX-N-2E\" transform=\"translate(500,0)\"><\/use><use data-c=\"34\" xlink:href=\"#MJX-5657-TEX-N-34\" transform=\"translate(778,0)\"><\/use><use data-c=\"38\" xlink:href=\"#MJX-5657-TEX-N-38\" transform=\"translate(1278,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14737.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5657-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo>:<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo stretchy=\"false\">\u21a6<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><\/mrow><\/msub><mo>+<\/mo><msub><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><mo>,<\/mo><mtext mathvariant=\"bold\">X<\/mtext><\/mrow><\/msub><msubsup><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mtext mathvariant=\"bold\">X<\/mtext><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">x<\/mi><mo>\u2212<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold-italic\">x<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mo stretchy=\"false\">(<\/mo><mn>5.48<\/mn><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The respective MMSE is equal to the trace of the conditional covariance matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.333ex\" height=\"2.375ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 2357.2 1049.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.798ex;\"><defs><path id=\"MJX-227-TEX-B-1D402\" d=\"M64 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transform=\"translate(894,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-227-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(1172,0)\"><use data-c=\"1D417\" xlink:href=\"#MJX-227-TEX-B-1D417\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mtext mathvariant=\"bold\">C<\/mtext><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mtext mathvariant=\"bold\">X<\/mtext><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\textbf{C}_{\\bold\\Theta|\\textbf{X}}<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n666\" cid=\"n666\" mdtype=\"math_block\" data-math-tag-before=\"1\" 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data-mjx-texclass=\"ORD\"><mtext mathvariant=\"bold\">X<\/mtext><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u0398<\/mi><\/mrow><\/mrow><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>Yes, the estimator in (5.48) is the solution of the integral in (5.34), where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-228-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-228-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\u03b8<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4206-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> are substituted by the respective vectors. The derivation of the estimator is however not as straightforward than for the scalar case due to the different rules for operations of vectors and matrices instead of scalars. <\/span><\/p><p><span>If you refer to the last paragraph on slide <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.262ex\" height=\"1.579ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -676 1000 698\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-234-TEX-N-37\" d=\"M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z\"><\/path><path id=\"MJX-234-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"37\" xlink:href=\"#MJX-234-TEX-N-37\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-234-TEX-N-35\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>75<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">75<\/script><span>, you find the case for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.157ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2721.6 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-235-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-235-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-235-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-235-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1165.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-235-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2221.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-235-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N=1<\/script><span>. There, if you consider <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.105ex\" height=\"2.565ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 2256.5 1133.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.679ex;\"><defs><path id=\"MJX-232-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-232-TEX-N-2F\" d=\"M423 750Q432 750 438 744T444 730Q444 725 271 248T92 -240Q85 -250 75 -250Q68 -250 62 -245T56 -231Q56 -221 230 257T407 740Q411 750 423 750Z\"><\/path><path id=\"MJX-232-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-232-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-232-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"31\" xlink:href=\"#MJX-232-TEX-N-31\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(500,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-232-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"msubsup\" transform=\"translate(1000,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-232-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-232-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-300) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-232-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msubsup><mi>\u03c3<\/mi><mi>X<\/mi><mn>2<\/mn><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">1\/\u03c3^2_X<\/script><span> to be equal with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.07ex\" height=\"2.697ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -882.5 1798.9 1191.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.7ex;\"><defs><path id=\"MJX-233-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-233-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-233-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-233-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msubsup\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-233-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(845.3,411.6) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-233-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-233-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(748,-309.4) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-233-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msubsup><mi>C<\/mi><mi>X<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">C^{\u22121}_X<\/script><span>, you certainly see the strong similarity to the Eq. (5.48).<\/span><\/p><p><span>Silde <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.262ex\" height=\"1.579ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -676 1000 698\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-234-TEX-N-37\" d=\"M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z\"><\/path><path id=\"MJX-234-TEX-N-35\" d=\"M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 420Q337 420 393 355T449 201Q449 109 385 44T229 -22Q148 -22 99 32T50 154Q50 178 61 192T84 210T107 214Q132 214 148 197T164 157Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mn\"><use data-c=\"37\" xlink:href=\"#MJX-234-TEX-N-37\"><\/use><use data-c=\"35\" xlink:href=\"#MJX-234-TEX-N-35\" transform=\"translate(500,0)\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mn>75<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">75<\/script><span>: For <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"6.157ex\" height=\"1.731ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2721.6 765\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.186ex;\"><defs><path id=\"MJX-235-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-235-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-235-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-235-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1165.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-235-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(2221.6,0)\"><use data-c=\"31\" xlink:href=\"#MJX-235-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N=1<\/script><span>, we obtain<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n672\" cid=\"n672\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 53.314ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"15.165ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -7.017ex; min-width: 53.314ex;\"><defs><path id=\"MJX-5659-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-5659-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 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mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/mrow><\/mfrac><mo>=<\/mo><mfrac><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mi>x<\/mi><mo>+<\/mo><mi>m<\/mi><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mtd><\/mtr><mtr><mtd><mo>=<\/mo><mi>m<\/mi><mo>+<\/mo><mfrac><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mrow><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><\/mrow><mn>2<\/mn><\/msubsup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>X<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>=<\/mo><mi>\u03b8<\/mi><\/mrow><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>m<\/mi><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><\/li><li><p><span>Given <\/span><strong><span>Jointly Gaussian Random Variables<\/span><\/strong><span> <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 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637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44C\" xlink:href=\"#MJX-237-TEX-I-1D44C\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>Y<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">Y<\/script><span>, the Conditional Mean Estimator <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"7.269ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3213 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-238-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 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data-mml-node=\"mi\"><use data-c=\"1D438\" xlink:href=\"#MJX-238-TEX-I-1D438\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(764,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-238-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1042,0)\"><use data-c=\"1D44C\" xlink:href=\"#MJX-238-TEX-I-1D44C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1805,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-238-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2083,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-238-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2935,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-238-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi>Y<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mi>X<\/mi><mo stretchy=\"false\">]<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">E [Y |X ]<\/script><span> is a Linear Function in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4206-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> (<\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"5.486ex\" height=\"2.034ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 2424.8 899\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.489ex;\"><defs><path id=\"MJX-2971-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-2971-TEX-I-1D70C\" d=\"M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z\"><\/path><path id=\"MJX-2971-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mo\"><use data-c=\"3D\" xlink:href=\"#MJX-2971-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1055.8,0)\"><use data-c=\"1D70C\" xlink:href=\"#MJX-2971-TEX-I-1D70C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1572.8,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-2971-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mo>=<\/mo><mi>\u03c1<\/mi><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">=\\rho X<\/script><span>), and the marginal distribution of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4206-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span> is also Gaussian independent of the correlation coefficient. This does not hold for arbitrarily jointly distributed random variables!<\/span><\/p><\/li><\/ul><h4 id='example-case'><span>Example case:<\/span><\/h4><p><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.556ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1130 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-2977-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-2977-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2977-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(836,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-2977-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>N<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N_i<\/script><span> are i.i.d. and follow <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.013ex\" height=\"2.565ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 3983.6 1133.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.679ex;\"><defs><path id=\"MJX-2979-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-2979-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-2979-TEX-N-30\" d=\"M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z\"><\/path><path id=\"MJX-2979-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-2979-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-2979-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-2979-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-2979-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-2979-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(979,0)\"><use data-c=\"28\" xlink:href=\"#MJX-2979-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1368,0)\"><use data-c=\"30\" xlink:href=\"#MJX-2979-TEX-N-30\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1868,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-2979-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2312.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-2979-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,363) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-2979-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-300) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-2979-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3594.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-2979-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mn>0<\/mn><mo>,<\/mo><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal{N}(0,\\sigma_N^2)<\/script><span>. Noise <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.061ex\" height=\"1.618ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 469 715\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.023ex;\"><defs><path id=\"MJX-2980-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-2980-TEX-I-1D703\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b8<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\theta<\/script><span> is the same for each signal <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.613ex\" height=\"1.902ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1155 840.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-2983-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-2983-TEX-I-1D456\" d=\"M184 600Q184 624 203 642T247 661Q265 661 277 649T290 619Q290 596 270 577T226 557Q211 557 198 567T184 600ZM21 287Q21 295 30 318T54 369T98 420T158 442Q197 442 223 419T250 357Q250 340 236 301T196 196T154 83Q149 61 149 51Q149 26 166 26Q175 26 185 29T208 43T235 78T260 137Q263 149 265 151T282 153Q302 153 302 143Q302 135 293 112T268 61T223 11T161 -11Q129 -11 102 10T74 74Q74 91 79 106T122 220Q160 321 166 341T173 380Q173 404 156 404H154Q124 404 99 371T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-2983-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-2983-TEX-I-1D456\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>X<\/mi><mi>i<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X_i<\/script><span>. We have<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n877\" cid=\"n877\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-labels=\"[]\" data-math-tag-after=\"1\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"41.922ex\" height=\"9.842ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -2425 18529.7 4350\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -4.355ex;\"><defs><path id=\"MJX-5660-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-5660-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A1\" d=\"M319 -645V1154H666V1070H403V-645H319Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A3\" d=\"M319 -644V1155H403V-560H666V-644H319Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A2\" d=\"M319 0V602H403V0H319Z\"><\/path><path id=\"MJX-5660-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5660-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-5660-TEX-N-22EE\" d=\"M78 30Q78 54 95 72T138 90Q162 90 180 74T199 31Q199 6 182 -12T139 -30T96 -13T78 30ZM78 440Q78 464 95 482T138 500Q162 500 180 484T199 441Q199 416 182 398T139 380T96 397T78 440ZM78 840Q78 864 95 882T138 900Q162 900 180 884T199 841Q199 816 182 798T139 780T96 797T78 840Z\"><\/path><path id=\"MJX-5660-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A4\" d=\"M0 1070V1154H347V-645H263V1070H0Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A6\" d=\"M263 -560V1155H347V-644H0V-560H263Z\"><\/path><path id=\"MJX-5660-TEX-S4-23A5\" d=\"M263 0V602H347V0H263Z\"><\/path><path id=\"MJX-5660-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-5660-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-5660-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5660-TEX-I-1D707\" d=\"M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z\"><\/path><path id=\"MJX-5660-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-5660-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-5660-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5660-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-5660-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-5660-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5660-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z\"><\/path><path id=\"MJX-5660-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5660-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1146.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5660-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mrow\" transform=\"translate(2202.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"23A1\" xlink:href=\"#MJX-5660-TEX-S4-23A1\" transform=\"translate(0,1271)\"><\/use><use data-c=\"23A3\" xlink:href=\"#MJX-5660-TEX-S4-23A3\" transform=\"translate(0,-1281)\"><\/use><svg width=\"667\" height=\"952\" y=\"-226\" x=\"0\" viewBox=\"0 238 667 952\"><use data-c=\"23A2\" xlink:href=\"#MJX-5660-TEX-S4-23A2\" transform=\"scale(1,2.372)\"><\/use><\/svg><\/g><g data-mml-node=\"mtable\" transform=\"translate(667,0)\"><g data-mml-node=\"mtr\" transform=\"translate(0,1675)\"><g data-mml-node=\"mtd\" transform=\"translate(137.2,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5660-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5660-TEX-N-31\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-275)\"><g data-mml-node=\"mtd\" transform=\"translate(630.5,0)\"><g data-mml-node=\"TeXAtom\" 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xlink:href=\"#MJX-5660-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13190.1,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-5660-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(14245.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-5660-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(15224.9,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5660-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(15613.9,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5660-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16491.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5660-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(16936.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5660-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5660-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-265.5) scale(0.707)\"><use data-c=\"398\" xlink:href=\"#MJX-5660-TEX-N-398\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(18140.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5660-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right\" columnspacing=\"\" rowspacing=\"3pt\"><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnalign=\"center\" columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi>X<\/mi><mn>1<\/mn><\/msub><\/mtd><\/mtr><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mo>\u22ee<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd><msub><mi>X<\/mi><mi>N<\/mi><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi>\u03bc<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mrow><\/msub><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi mathvariant=\"normal\">\u0398<\/mi><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo 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data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mi>m<\/mi><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">N<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><msup><mo stretchy=\"false\">]<\/mo><mn>2<\/mn><\/msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><msup><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msup><mo stretchy=\"false\">]<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><msup><mo stretchy=\"false\">]<\/mo><mn>2<\/mn><\/msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mi>C<\/mi><mi>o<\/mi><mi>v<\/mi><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">N<\/mi><\/mrow><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">N<\/mi><\/mrow><mo>\u2212<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">]<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo>+<\/mo><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">I<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Apply (5.48), we have the CM estimator<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n887\" cid=\"n887\" mdtype=\"math_block\" 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mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='55-orthogonality-principle'><span>5.5 Orthogonality Principle<\/span><\/h3><p><span>The <\/span><strong><span>stochastic orthogonality<\/span><\/strong><span> is an inherent property of the conditional mean estimator, it describes the inherent stochastic orthogonality between the CM estimator error and <\/span><strong><span>any<\/span><\/strong><span> observations statistics thereof:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n679\" cid=\"n679\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 49.037ex; position: 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transform=\"translate(17087.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5663-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(17476.5,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5663-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(17726.5,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5663-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(17976.5,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5663-TEX-N-A0\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2250 1 4000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d4<\/mo><\/mtd><\/mtr><mtr><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>\u22a5<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>the CM estimation error is stochastically orthogonal to any observations statistics.<\/span><img decoding=\"async\" src=\".\/Statistical Signal Processing.assets\/\u622a\u5c4f2023-06-17 22.57.10.png\" referrerpolicy=\"no-referrer\" alt=\"\u622a\u5c4f2023-06-17 22.57.10\"><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.912ex\" height=\"1.654ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 845 731\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.109ex;\"><defs><path id=\"MJX-4221-TEX-C-48\" d=\"M18 487Q18 496 29 517T67 566T127 621T216 665T330 683Q359 683 376 669T397 643T400 622Q400 584 382 488T348 343Q348 342 467 342H587L594 366Q615 440 648 534T690 641Q701 656 723 669T764 683Q783 683 783 672L750 578Q716 485 677 346T625 101Q624 92 623 82T622 65T621 56Q621 20 658 20Q666 20 701 25Q709 52 736 69T785 87Q803 87 803 75T791 44T754 3T685 -33T588 -48Q568 -48 562 -46Q522 -31 522 13V23Q531 129 562 250L569 281L565 280Q561 278 556 277T549 274L438 273H328L321 249Q307 202 275 107T232 0Q219 -16 196 -28T155 -41Q149 -41 145 -39T140 -34T139 -29Q139 -24 148 -3T181 86T233 247Q240 270 240 272Q240 273 194 273H169Q139 273 139 285Q139 295 153 308T187 332Q206 341 236 342L260 343L264 359Q278 414 289 482T300 578Q300 613 260 613H254Q198 613 169 592Q148 578 127 544T104 508Q72 478 37 475Q18 475 18 487Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"48\" xlink:href=\"#MJX-4221-TEX-C-48\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal H<\/script><span> is the subspace spanned by a random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.991ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 2206 1000\" 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xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h(X)<\/script><span> which is based on the random variable <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.928ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 852 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-4206-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4206-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">X<\/script><span>, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.094ex\" height=\"2.262ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -750 3577.6 1000\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-4207-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-4207-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-4207-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 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id=\"MJX-4207-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-4207-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-4207-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-4207-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-4207-TEX-I-1D440\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1947.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-4207-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2336.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4207-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3188.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-4207-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_{CM}(X)<\/script><span> is the CM estimator. The error <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.004ex\" height=\"1.97ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 5747.6 870.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.375ex;\"><defs><path id=\"MJX-4219-TEX-I-1D700\" d=\"M190 -22Q124 -22 76 11T27 107Q27 174 97 232L107 239L99 248Q76 273 76 304Q76 364 144 408T290 452H302Q360 452 405 421Q428 405 428 392Q428 381 417 369T391 356Q382 356 371 365T338 383T283 392Q217 392 167 368T116 308Q116 289 133 272Q142 263 145 262T157 264Q188 278 238 278H243Q308 278 308 247Q308 206 223 206Q177 206 142 219L132 212Q68 169 68 112Q68 39 201 39Q253 39 286 49T328 72T345 94T362 105Q376 103 376 88Q376 79 365 62T334 26T275 -8T190 -22Z\"><\/path><path id=\"MJX-4219-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-4219-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-4219-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-4219-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-4219-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-4219-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D700\" xlink:href=\"#MJX-4219-TEX-I-1D700\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(743.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-4219-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1799.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-4219-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-4219-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-4219-TEX-I-1D440\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3969.3,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-4219-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4969.6,0)\"><use data-c=\"398\" xlink:href=\"#MJX-4219-TEX-N-398\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>\u03b5<\/mi><mo>=<\/mo><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\varepsilon=T_{CM}-\\Theta<\/script><span> should be orthogonal to <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.912ex\" height=\"1.654ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 845 731\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.109ex;\"><defs><path id=\"MJX-4221-TEX-C-48\" d=\"M18 487Q18 496 29 517T67 566T127 621T216 665T330 683Q359 683 376 669T397 643T400 622Q400 584 382 488T348 343Q348 342 467 342H587L594 366Q615 440 648 534T690 641Q701 656 723 669T764 683Q783 683 783 672L750 578Q716 485 677 346T625 101Q624 92 623 82T622 65T621 56Q621 20 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data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\mathcal H<\/script><span> to be minimal.<\/span><\/p><h4 id='mean-estimation-example-2'><span>Mean Estimation Example<\/span><\/h4><p><span>Given<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n683\" cid=\"n683\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 41.991ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"6.785ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.827ex; min-width: 41.991ex;\"><defs><path id=\"MJX-5664-TEX-S3-5B\" d=\"M247 -949V1450H516V1388H309V-887H516V-949H247Z\"><\/path><path id=\"MJX-5664-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 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xlink:href=\"#MJX-5665-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(609,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5665-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1180.7,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5665-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2236.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5665-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-5665-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3613.7,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5665-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4613.9,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D707\" xlink:href=\"#MJX-5665-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(636,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5665-TEX-I-1D44B\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>h<\/mi><mi>i<\/mi><\/msub><mo>=<\/mo><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Recall that the MSE optimal estimator is linear in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.143ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 947 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-242-TEX-BI-1D47F\" d=\"M931 686Q953 686 953 670Q953 650 944 632Q936 624 924 624H914Q823 624 803 611Q800 609 696 503T591 396Q591 394 667 229L743 62H787H814Q846 62 846 44Q843 7 829 2Q825 0 817 0Q813 0 775 1T664 2Q590 2 551 1T508 0H507Q484 0 484 18Q484 19 488 37Q492 56 497 58T534 62L566 63Q567 64 520 169T471 274Q469 274 369 172T268 67L315 62Q320 62 328 62L335 61Q347 58 347 44Q344 10 331 2L326 0L287 1Q263 2 177 2Q95 2 78 1L53 0Q38 6 38 17Q38 40 50 57Q56 62 78 62Q169 62 188 75Q194 77 435 324L444 334L439 347Q437 351 373 492L313 624H268H246Q220 624 212 632Q210 636 210 642Q210 655 215 669T227 684Q230 686 247 686Q295 684 398 684Q438 684 472 684T527 685T551 686Q567 686 572 671Q572 667 568 651Q563 631 558 628T523 624T492 623H488L526 540Q563 457 564 457Q564 456 574 466T604 496T645 537L724 619Q716 622 677 624H673Q645 624 645 640Q645 660 654 678Q659 683 666 686L704 685Q728 684 813 684Q847 684 873 684T913 685T931 686Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D47F\" xlink:href=\"#MJX-242-TEX-BI-1D47F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi mathvariant=\"bold-italic\">X<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\boldsymbol X<\/script><span> due to the jointly Gaussian distribution, we substitute <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.406ex\" height=\"1.906ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -677 1947.6 842.6\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 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data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">T_{CM}<\/script><span> by a <\/span><strong><span>linear model<\/span><\/strong><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n687\" cid=\"n687\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 23.534ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.601ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.735ex; min-width: 23.534ex;\"><defs><path id=\"MJX-5666-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 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data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5666-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(6203.7,0)\"><use data-c=\"1D47F\" xlink:href=\"#MJX-5666-TEX-BI-1D47F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7372.9,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5666-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8373.1,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-5666-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -824.7 1 1149.5\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-74.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msup><mi mathvariant=\"bold-italic\">a<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mi mathvariant=\"bold-italic\">X<\/mi><mo>+<\/mo><mi>b<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Apply the orthogonal principle,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n895\" cid=\"n895\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-labels=\"[]\" data-math-tag-after=\"1\"><div class=\"md-rawblock-container md-math-container\" 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data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">)<\/mo><mi>h<\/mi><mo stretchy=\"false\">(<\/mo><msub><mi>X<\/mi><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mi>X<\/mi><mi>N<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>we obtain<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n689\" cid=\"n689\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" 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xlink:href=\"#MJX-5668-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\" columnalign=\"right left\" columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi mathvariant=\"bold-italic\">a<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mi mathvariant=\"bold-italic\">X<\/mi><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">X<\/mi><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><mtd><mi><\/mi><mo>=<\/mo><mi>E<\/mi><mo stretchy=\"false\">[<\/mo><mo stretchy=\"false\">(<\/mo><msup><mi mathvariant=\"bold-italic\">a<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">X<\/mi><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mi>b<\/mi><mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"normal\">\u0398<\/mi><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">X<\/mi><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><mo>\u2212<\/mo><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The missing parameter <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"0.971ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 429 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-244-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-244-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>b<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">b<\/script><span> is eventually found by introducing another function <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"9.104ex\" height=\"2.041ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 4024.1 902\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.471ex;\"><defs><path id=\"MJX-245-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-245-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-245-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-245-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-245-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-245-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-245-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(888,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-245-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(1666,0)\"><use data-c=\"31\" xlink:href=\"#MJX-245-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2468.4,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-245-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(3524.1,0)\"><use data-c=\"31\" xlink:href=\"#MJX-245-TEX-N-31\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>N<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msub><mo>=<\/mo><mn>1<\/mn><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_{N+1}=1<\/script><span>:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n691\" cid=\"n691\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 21.688ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.601ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.735ex; min-width: 21.688ex;\"><defs><path id=\"MJX-5669-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><path id=\"MJX-5669-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5669-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5669-TEX-I-1D707\" d=\"M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z\"><\/path><path id=\"MJX-5669-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5669-TEX-B-1D41A\" d=\"M64 349Q64 399 107 426T255 453Q346 453 402 423T473 341Q478 327 478 310T479 196V77Q493 63 529 62Q549 62 553 57T558 31Q558 9 552 5T514 0H497H481Q375 0 367 56L356 46Q300 -6 210 -6Q130 -6 81 30T32 121Q32 188 111 226T332 272H350V292Q350 313 348 327T337 361T306 391T248 402T194 399H189Q204 376 204 354Q204 327 187 306T134 284Q97 284 81 305T64 349ZM164 121Q164 89 186 67T238 45Q274 45 307 63T346 108L350 117V226H347Q248 218 206 189T164 121Z\"><\/path><path id=\"MJX-5669-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-5669-TEX-B-1D7CF\" d=\"M481 0L294 3Q136 3 109 0H96V62H227V304Q227 546 225 546Q169 529 97 529H80V591H97Q231 591 308 647L319 655H333Q355 655 359 644Q361 640 361 351V62H494V0H481Z\"><\/path><path id=\"MJX-5669-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5669-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -824.7)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 824.7) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3993.1 -824.7 1 1149.5\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-74.7)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mi\"><use data-c=\"1D44F\" xlink:href=\"#MJX-5669-TEX-I-1D44F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(706.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5669-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1762.6,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5669-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2540.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D707\" xlink:href=\"#MJX-5669-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(636,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5669-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(3829,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D41A\" xlink:href=\"#MJX-5669-TEX-B-1D41A\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(592,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5669-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4952.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CF\" xlink:href=\"#MJX-5669-TEX-B-1D7CF\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5749.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5669-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(6750,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D707\" xlink:href=\"#MJX-5669-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(636,-150) scale(0.707)\"><use data-c=\"398\" xlink:href=\"#MJX-5669-TEX-N-398\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -824.7 1 1149.5\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-74.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">a<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo>+<\/mo><msub><mi>\u03bc<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Finally we get<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n928\" cid=\"n928\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-labels=\"[]\" data-math-tag-after=\"1\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 39.999ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.65ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.759ex; min-width: 39.999ex;\"><defs><path id=\"MJX-5670-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-5670-TEX-I-1D436\" d=\"M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z\"><\/path><path id=\"MJX-5670-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-5670-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5670-TEX-I-1D44B\" d=\"M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 46 734 37Q734 35 732 23Q728 7 725 4T711 1Q708 1 678 1T589 2Q528 2 496 2T461 1Q444 1 444 10Q444 11 446 25Q448 35 450 39T455 44T464 46T480 47T506 54Q523 62 523 64Q522 64 476 181L429 299Q241 95 236 84Q232 76 232 72Q232 53 261 47Q262 47 267 47T273 46Q276 46 277 46T280 45T283 42T284 35Q284 26 282 19Q279 6 276 4T261 1Q258 1 243 1T201 2T142 2Q64 2 42 0Z\"><\/path><path id=\"MJX-5670-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5670-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5670-TEX-N-398\" d=\"M56 340Q56 423 86 494T164 610T270 680T388 705Q521 705 621 601T722 341Q722 260 693 191T617 75T510 4T388 -22T267 3T160 74T85 189T56 340ZM610 339Q610 428 590 495T535 598T463 651T384 668Q332 668 289 638T221 566Q168 485 168 339Q168 274 176 235Q189 158 228 105T324 28Q356 16 388 16Q415 16 442 24T501 54T555 111T594 205T610 339ZM223 263V422H263V388H514V422H554V263H514V297H263V263H223Z\"><\/path><path id=\"MJX-5670-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-5670-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-5670-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-5670-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5670-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-5670-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-5670-TEX-I-1D707\" d=\"M58 -216Q44 -216 34 -208T23 -186Q23 -176 96 116T173 414Q186 442 219 442Q231 441 239 435T249 423T251 413Q251 401 220 279T187 142Q185 131 185 107V99Q185 26 252 26Q261 26 270 27T287 31T302 38T315 45T327 55T338 65T348 77T356 88T365 100L372 110L408 253Q444 395 448 404Q461 431 491 431Q504 431 512 424T523 412T525 402L449 84Q448 79 448 68Q448 43 455 35T476 26Q485 27 496 35Q517 55 537 131Q543 151 547 152Q549 153 557 153H561Q580 153 580 144Q580 138 575 117T555 63T523 13Q510 0 491 -8Q483 -10 467 -10Q446 -10 429 -4T402 11T385 29T376 44T374 51L368 45Q362 39 350 30T324 12T288 -4T246 -11Q199 -11 153 12L129 -85Q108 -167 104 -180T92 -202Q76 -216 58 -216Z\"><\/path><path id=\"MJX-5670-TEX-B-1D7CF\" d=\"M481 0L294 3Q136 3 109 0H96V62H227V304Q227 546 225 546Q169 529 97 529H80V591H97Q231 591 308 647L319 655H333Q355 655 359 644Q361 640 361 351V62H494V0H481Z\"><\/path><path id=\"MJX-5670-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -835.6)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 835.6) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"8039.7 -835.6 1 1171.1\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-48.4)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-5670-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-5670-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5670-TEX-I-1D440\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1947.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5670-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2336.6,0)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5670-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3188.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5670-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3855.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5670-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4911.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-5670-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(748,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"398\" xlink:href=\"#MJX-5670-TEX-N-398\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(778,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5670-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1056,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5670-TEX-B-1D417\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"msubsup\" transform=\"translate(7070.3,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-5670-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-5670-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5670-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-247) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-5670-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8888,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5670-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9277,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5670-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10368.2,0)\"><use data-c=\"2212\" xlink:href=\"#MJX-5670-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(11368.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D707\" xlink:href=\"#MJX-5670-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(636,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-5670-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(12656.9,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CF\" xlink:href=\"#MJX-5670-TEX-B-1D7CF\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13231.9,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5670-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13843.1,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5670-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(14843.3,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D707\" xlink:href=\"#MJX-5670-TEX-I-1D707\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(636,-150) scale(0.707)\"><use data-c=\"398\" xlink:href=\"#MJX-5670-TEX-N-398\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -835.6 1 1171.1\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-48.4)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi>X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><msub><mi>C<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"normal\">\u0398<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo>\u2212<\/mo><msub><mi>\u03bc<\/mi><mi>X<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">1<\/mn><\/mrow><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>\u03bc<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><ul><li><p><span>Alternative solution: <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.34ex\" height=\"2.767ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1073 3686.3 1223\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-4883-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-4883-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-4883-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-4883-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-4883-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-4883-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 336Q297 271 279 198T252 88L243 52Q243 48 252 48T311 46H328Q360 46 379 47T428 54T478 72T522 106T564 161Q580 191 594 228T611 270Q616 273 628 273H641Q647 264 647 262T627 203T583 83T557 9Q555 4 553 3T537 0T494 -1Q483 -1 418 -1T294 0H116Q32 0 32 10Q32 17 34 24Q39 43 44 45Q48 46 59 46H65Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Q285 635 228 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-4883-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(853.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-4883-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1909.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D703\" xlink:href=\"#MJX-4883-TEX-I-1D703\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(317.8,279) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-4883-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(502,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D440\" xlink:href=\"#MJX-4883-TEX-I-1D440\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1051,0)\"><use data-c=\"1D43F\" xlink:href=\"#MJX-4883-TEX-I-1D43F\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>h<\/mi><mo>=<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h=\\hat\\theta_{ML}<\/script><\/p><\/li><\/ul><p><span>We choose <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"27.137ex\" height=\"3.204ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -1073 11994.6 1416.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.777ex;\"><defs><path id=\"MJX-4885-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-4885-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-4885-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-4885-TEX-I-1D703\" d=\"M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z\"><\/path><path id=\"MJX-4885-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-4885-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 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data-c=\"31\" xlink:href=\"#MJX-4885-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(9451.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D44B\" xlink:href=\"#MJX-4885-TEX-I-1D44B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(861,-150) scale(0.707)\"><use data-c=\"1D456\" xlink:href=\"#MJX-4885-TEX-I-1D456\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10606.6,0)\"><g data-mml-node=\"mo\"><use data-c=\"2F\" xlink:href=\"#MJX-4885-TEX-N-2F\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(11106.6,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-4885-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>h<\/mi><mo>=<\/mo><mi>S<\/mi><mo>=<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mover><mi>\u03b8<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>M<\/mi><mi>L<\/mi><\/mrow><\/msub><mo>=<\/mo><munderover><mo data-mjx-texclass=\"OP\">\u2211<\/mo><mrow data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>X<\/mi><mi>i<\/mi><\/msub><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h=S=\\hat\\theta_{ML}=\\sum_{i=1}^NX_i\/N<\/script><span>, and the MSE optimal estimator is linear in <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.143ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 947 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-242-TEX-BI-1D47F\" d=\"M931 686Q953 686 953 670Q953 650 944 632Q936 624 924 624H914Q823 624 803 611Q800 609 696 503T591 396Q591 394 667 229L743 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214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z\"><\/path><path id=\"MJX-5671-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><path id=\"MJX-5671-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5671-TEX-I-1D44F\" d=\"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3915.8 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D447\" xlink:href=\"#MJX-5671-TEX-I-1D447\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(617,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D436\" xlink:href=\"#MJX-5671-TEX-I-1D436\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(760,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5671-TEX-I-1D440\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1947.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5671-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(2336.6,0)\"><use data-c=\"1D47F\" xlink:href=\"#MJX-5671-TEX-BI-1D47F\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3283.6,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5671-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3950.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5671-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5006.1,0)\"><use data-c=\"1D44E\" xlink:href=\"#MJX-5671-TEX-I-1D44E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(5535.1,0)\"><use data-c=\"1D446\" xlink:href=\"#MJX-5671-TEX-I-1D446\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6402.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5671-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7402.6,0)\"><use data-c=\"1D44F\" xlink:href=\"#MJX-5671-TEX-I-1D44F\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>C<\/mi><mi>M<\/mi><\/mrow><\/msub><mo stretchy=\"false\">(<\/mo><mi mathvariant=\"bold-italic\">X<\/mi><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>a<\/mi><mi>S<\/mi><mo>+<\/mo><mi>b<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>and apply orthogonality principle:<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n905\" cid=\"n905\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-labels=\"[]\" data-math-tag-after=\"1\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 34.893ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"16.295ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -7.582ex; min-width: 34.893ex;\"><defs><path id=\"MJX-5672-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 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mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msubsup><\/mrow><mrow><mi>N<\/mi><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msubsup><mi>\u03c3<\/mi><mi>N<\/mi><mn>2<\/mn><\/msubsup><mo>+<\/mo><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mo>\/<\/mo><\/mrow><msubsup><mi>\u03c3<\/mi><mi mathvariant=\"normal\">\u0398<\/mi><mn>2<\/mn><\/msubsup><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p>&nbsp;<\/p><p>&nbsp;<\/p><h1 id='iv-linear-estimation'><span>IV. Linear Estimation<\/span><\/h1><h2 id='6-linear-estimation'><span>6. 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mathvariant=\"bold\">X<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mn>1<\/mn><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><\/mtd><\/mtr><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mo>\u22ee<\/mo><\/mrow><\/mtd><\/mtr><mtr><mtd><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mi>N<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='61-least-square-estimation'><span>6.1 Least Square Estimation<\/span><\/h3><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msubsup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mn>2<\/mn><mn>2<\/mn><\/msubsup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>The standard approach for this is based on convexity of the objective function,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n702\" cid=\"n702\" mdtype=\"math_block\" data-math-tag-before=\"1\" data-math-tag-after=\"1\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 40.558ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"8.481ex\" role=\"img\" focusable=\"false\" 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mathvariant=\"bold\">t<\/mi><\/mrow><mo>\u2208<\/mo><mtext>null<\/mtext><mo stretchy=\"false\">[<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><mtext>&nbsp;<\/mtext><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><mi>S<\/mi><\/mrow><\/msub><mo>=<\/mo><mo stretchy=\"false\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='2nd-geometrical-perspective'><span>2nd Geometrical Perspective<\/span><\/h4><p><span>Using orthogonal projector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"14.411ex\" height=\"2.488ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 6369.8 1099.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-246-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-246-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-246-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-246-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-246-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-246-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-246-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(869,0)\"><use data-c=\"28\" xlink:href=\"#MJX-246-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(1258,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-246-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(902,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-246-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2691.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-246-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(3560.5,0)\"><g data-mml-node=\"mo\"><use data-c=\"29\" xlink:href=\"#MJX-246-TEX-N-29\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(422,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-246-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-246-TEX-N-31\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(4936.2,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-246-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(902,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-246-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold X(\\bold X^\\mathsf{T}\\bold X)^{-1}\\bold X^\\mathsf{T}<\/script><span>, we project <\/span><mjx-container 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data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><msup><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='mean-estimation-example-3'><span>Mean Estimation Example<\/span><\/h4><p><span>In order to estimate the mean <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.915ex\" height=\"1.489ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 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data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D453\" xlink:href=\"#MJX-250-TEX-I-1D453\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(523,-150) scale(0.707)\"><use data-c=\"1D44B\" xlink:href=\"#MJX-250-TEX-I-1D44B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1175.5,0)\"><use data-c=\"28\" xlink:href=\"#MJX-250-TEX-N-28\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1564.5,0)\"><use data-c=\"1D465\" xlink:href=\"#MJX-250-TEX-I-1D465\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(2136.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-250-TEX-N-29\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>f<\/mi><mi>X<\/mi><\/msub><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">f_X(x)<\/script><span> based on <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" 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data-mjx-texclass=\"ORD\"><mi>i<\/mi><mo>=<\/mo><mn>1<\/mn><\/mrow><mi>N<\/mi><\/munderover><msub><mi>x<\/mi><mi>i<\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='affine-linear-regression'><span>Affine Linear Regression<\/span><\/h4><p><span>Given traning set, a linear estimator is defined as <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n716\" cid=\"n716\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 20.666ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"8.645ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.757ex; min-width: 20.666ex;\"><defs><path id=\"MJX-5741-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z\"><\/path><path id=\"MJX-5741-TEX-I-1D43F\" d=\"M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 217 683Q271 680 344 680Q485 680 506 683H518Q524 677 524 674T522 656Q517 641 513 637H475Q406 636 394 628Q387 624 380 600T313 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xlink:href=\"#MJX-5741-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mfrac\" transform=\"translate(3993,0)\"><g data-mml-node=\"mrow\" transform=\"translate(220,676)\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-5741-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(640,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5741-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1171.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5741-TEX-B-1D432\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mrow\" transform=\"translate(220,-735.5)\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-5741-TEX-B-1D431\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(640,289) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5741-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1171.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-5741-TEX-B-1D431\"><\/use><\/g><\/g><\/g><rect width=\"1978.5\" height=\"60\" x=\"120\" y=\"220\"><\/rect><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -2160.5 1 3821\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msub><mi>T<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><mi>i<\/mi><mi>n<\/mi><\/mrow><\/msub><mo>:<\/mo><mi>x<\/mi><mo stretchy=\"false\">\u21a6<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><mi>y<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>=<\/mo><mi>x<\/mi><mi>t<\/mi><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msub><mi>t<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>L<\/mi><mi>S<\/mi><\/mrow><\/msub><mo>=<\/mo><mfrac><mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/mrow><mrow><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/mfrac><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Another linear estimator is <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n944\" cid=\"n944\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-labels=\"[]\" data-math-tag-after=\"2\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 44.739ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"9.162ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -4.015ex; min-width: 44.739ex;\"><defs><path id=\"MJX-5795-TEX-I-1D447\" d=\"M40 437Q21 437 21 445Q21 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data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">u<\/mi><\/mrow><mi>i<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">v<\/mi><\/mrow><mi>i<\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h3 id='62-least-equares-estimation-with-regularization-ridge-regression'><span>6.2 Least Equares Estimation with Regularization (Ridge Regression)<\/span><\/h3><p><span>In many real-world applications, LS problems may be ill-conditioned. 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data-mjx-texclass=\"ORD\" transform=\"translate(5261.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5705-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5868.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(6146.4,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(311,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5705-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7083.2,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5705-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8083.4,0)\"><use data-c=\"1D6FE\" xlink:href=\"#MJX-5705-TEX-I-1D6FE\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8626.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8904.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9182.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D6BD\" xlink:href=\"#MJX-5705-TEX-B-1D6BD\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10013.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D42D\" xlink:href=\"#MJX-5705-TEX-B-1D42D\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10460.4,0) translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(10738.4,0)\"><g data-mml-node=\"mo\" transform=\"translate(0 -0.5)\"><use data-c=\"7C\" xlink:href=\"#MJX-5705-TEX-N-7C\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(311,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5705-TEX-N-32\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11453,0)\"><use data-c=\"7D\" xlink:href=\"#MJX-5705-TEX-N-7D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1050.7 1 1601.4\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,166.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><munder><mo data-mjx-texclass=\"OP\" movablelimits=\"true\">min<\/mo><mi>t<\/mi><\/munder><mo fence=\"false\" stretchy=\"false\">{<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mo>\u2212<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mn>2<\/mn><\/msup><mo>+<\/mo><mi>\u03b3<\/mi><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u03a6<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><msup><mo data-mjx-texclass=\"ORD\" stretchy=\"false\">|<\/mo><mn>2<\/mn><\/msup><mo fence=\"false\" stretchy=\"false\">}<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.88ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 831 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-252-TEX-B-1D6BD\" d=\"M609 0Q582 3 415 3T221 0H207V62H342V168L328 169Q193 180 117 241Q64 286 64 343T117 445Q193 506 328 517L342 518V624H207V686H221Q248 683 415 683T609 686H623V624H488V518L502 517Q637 506 713 445Q766 400 766 343T713 241Q637 180 502 169L488 168V62H623V0H609ZM342 219T342 343T340 467Q328 467 304 459Q277 451 261 439T237 409T228 378T226 343Q226 314 229 296T250 259T301 228Q331 219 341 219Q342 219 342 343ZM604 343Q604 365 602 379T591 413T560 446T503 464L489 467Q488 467 488 343T489 219Q499 219 529 228Q554 236 570 248T593 277T602 308T604 343Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6BD\" xlink:href=\"#MJX-252-TEX-B-1D6BD\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u03a6<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold \\Phi<\/script><span> is the approproately chosen regularization operator. It can be shown that the solution is<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n724\" cid=\"n724\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 29.323ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.954ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.911ex; min-width: 29.323ex;\"><defs><path id=\"MJX-5706-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5706-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-5706-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-5706-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5706-TEX-I-1D6FE\" d=\"M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z\"><\/path><path id=\"MJX-5706-TEX-B-1D6BD\" d=\"M609 0Q582 3 415 3T221 0H207V62H342V168L328 169Q193 180 117 241Q64 286 64 343T117 445Q193 506 328 517L342 518V624H207V686H221Q248 683 415 683T609 686H623V624H488V518L502 517Q637 506 713 445Q766 400 766 343T713 241Q637 180 502 169L488 168V62H623V0H609ZM342 219T342 343T340 467Q328 467 304 459Q277 451 261 439T237 409T228 378T226 343Q226 314 229 296T250 259T301 228Q331 219 341 219Q342 219 342 343ZM604 343Q604 365 602 379T591 413T560 446T503 464L489 467Q488 467 488 343T489 219Q499 219 529 228Q554 236 570 248T593 277T602 308T604 343Z\"><\/path><path id=\"MJX-5706-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5706-TEX-B-1D42D\" d=\"M272 49Q320 49 320 136V145V177H382V143Q382 106 380 99Q374 62 349 36T285 -2L272 -5H247Q173 -5 134 27Q109 46 102 74T94 160Q94 171 94 199T95 245V382H21V433H25Q58 433 90 456Q121 479 140 523T162 621V635H224V444H363V382H224V239V207V149Q224 98 228 81T249 55Q261 49 272 49Z\"><\/path><path id=\"MJX-5706-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-5706-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5706-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -902.9)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 902.9) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"5680.3 -902.9 1 1305.7\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-100.1)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"28\" xlink:href=\"#MJX-5706-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(389,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5706-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(902,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5706-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1822.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5706-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2913.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5706-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(3914,0)\"><use data-c=\"1D6FE\" xlink:href=\"#MJX-5706-TEX-I-1D6FE\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4457,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D6BD\" xlink:href=\"#MJX-5706-TEX-B-1D6BD\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5706-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5852.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D6BD\" xlink:href=\"#MJX-5706-TEX-B-1D6BD\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6683.5,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5706-TEX-N-29\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(7072.5,0)\"><g data-mml-node=\"mover\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42D\" xlink:href=\"#MJX-5706-TEX-B-1D42D\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(480,-150) scale(0.707)\"><use data-c=\"1D6FE\" xlink:href=\"#MJX-5706-TEX-I-1D6FE\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(457,209) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-5706-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8264.3,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5706-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(9320,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5706-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(902,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5706-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(10753.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5706-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -902.9 1 1305.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-100.1)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo>+<\/mo><mi>\u03b3<\/mi><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u03a6<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">\u03a6<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mover><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mi>\u03b3<\/mi><\/msub><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>=<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>Furthermore, <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n726\" cid=\"n726\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" 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id='63-linear-minimum-mean-square-error-estimation'><span>6.3 Linear Minimum Mean Square Error Estimation<\/span><\/h3><p><span>Based on a linear model for the estimator, the LMMSE minimizes the optimization problem<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n729\" cid=\"n729\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 27.048ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"7.683ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -3.276ex; min-width: 27.048ex;\"><defs><path id=\"MJX-5798-TEX-I-1D447\" d=\"M40 437Q21 437 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xlink:href=\"#MJX-5709-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(490,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5709-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(768,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-5709-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mtd\" transform=\"translate(2801.2,0)\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-5709-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(466,-150) scale(0.707)\"><use data-c=\"1D466\" xlink:href=\"#MJX-5709-TEX-I-1D466\"><\/use><\/g><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4504.5,0) translate(0 -0.5)\"><use data-c=\"5D\" xlink:href=\"#MJX-5709-TEX-S3-5D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1495 1 2489.9\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mi>z<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mi>\u03bc<\/mi><mi>y<\/mi><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mi>z<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"INNER\"><mo data-mjx-texclass=\"OPEN\">[<\/mo><mtable columnspacing=\"1em\" rowspacing=\"4pt\"><mtr><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><\/mtd><mtd><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>,<\/mo><mi>y<\/mi><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><\/mtd><mtd><msub><mi>c<\/mi><mi>y<\/mi><\/msub><\/mtd><\/mtr><\/mtable><mo data-mjx-texclass=\"CLOSE\">]<\/mo><\/mrow><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>we can get the LMMSE estimator by<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n733\" cid=\"n733\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" contenteditable=\"false\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"59.94ex\" height=\"40.424ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -9183.6 26493.6 17867.3\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -19.646ex;\"><defs><path id=\"MJX-5803-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 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mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><mo>+<\/mo><msubsup><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><msub><mi>\u03bc<\/mi><mi>y<\/mi><\/msub><mo>\u2212<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo stretchy=\"false\">(<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/msub><mo>+<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/msub><msubsup><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>m<\/mi><msubsup><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msubsup><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mover><mo>=<\/mo><mo>!<\/mo><\/mover><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mo stretchy=\"false\">\u21d2<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">t<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><\/mtd><\/mtr><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mover><mi>y<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>+<\/mo><msub><mi>\u03bc<\/mi><mi>y<\/mi><\/msub><mo>\u2212<\/mo><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><mi><\/mi><mo>=<\/mo><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mo>\u2212<\/mo><msub><mi mathvariant=\"bold-italic\">\u03bc<\/mi><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><msub><mi>\u03bc<\/mi><mi>y<\/mi><\/msub><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>! apply orthogonality principle<\/span><\/p><ul><li><p><span>In the case of zero-mean random variables, the LMMSE estimator is <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"12.979ex\" height=\"2.554ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -833.9 5736.5 1128.9\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.667ex;\"><defs><path id=\"MJX-254-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-254-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-254-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-254-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 121Q410 121 419 112T429 98T420 83T391 55T346 25T282 0T202 -11Q127 -11 81 37T34 159Z\"><\/path><path id=\"MJX-254-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-254-TEX-B-1D431\" d=\"M227 0Q212 3 121 3Q40 3 28 0H21V62H117L245 213L109 382H26V444H34Q49 441 143 441Q247 441 265 444H274V382H246L281 339Q315 297 316 297Q320 297 354 341L389 382H352V444H360Q375 441 466 441Q547 441 559 444H566V382H471L355 246L504 63L545 62H586V0H578Q563 3 469 3Q365 3 347 0H338V62H366Q366 63 326 112T285 163L198 63L217 62H235V0H227Z\"><\/path><path id=\"MJX-254-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-254-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-254-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-254-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(300.6,16) translate(-250 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-254-TEX-N-5E\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(767.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-254-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(1823.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D450\" xlink:href=\"#MJX-254-TEX-I-1D450\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(466,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D466\" xlink:href=\"#MJX-254-TEX-I-1D466\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(490,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-254-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(768,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-254-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"msubsup\" transform=\"translate(3311.8,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-254-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mo\"><use data-c=\"2212\" xlink:href=\"#MJX-254-TEX-N-2212\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(778,0)\"><use data-c=\"31\" xlink:href=\"#MJX-254-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-247) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-254-TEX-B-1D431\"><\/use><\/g><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(5129.5,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D431\" xlink:href=\"#MJX-254-TEX-B-1D431\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mover><mi>y<\/mi><mo stretchy=\"false\">^<\/mo><\/mover><\/mrow><mo>=<\/mo><msub><mi>c<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>y<\/mi><mo>,<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/mrow><\/msub><msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/msubsup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\hat y=c_{y,\\bold x}\\bold C_\\bold x^{-1}\\bold x<\/script><span>. The minimum MSE is <\/span><\/p><\/li><\/ul><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n738\" cid=\"n738\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 46.158ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.702ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.786ex; min-width: 46.158ex;\"><defs><path id=\"MJX-5711-TEX-I-1D438\" d=\"M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z\"><\/path><path id=\"MJX-5711-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-5711-TEX-N-7C\" d=\"M139 -249H137Q125 -249 119 -235V251L120 737Q130 750 139 750Q152 750 159 735V-235Q151 -249 141 -249H139Z\"><\/path><path id=\"MJX-5711-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5711-TEX-N-2212\" d=\"M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z\"><\/path><path id=\"MJX-5711-TEX-N-5E\" d=\"M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z\"><\/path><path id=\"MJX-5711-TEX-N-32\" d=\"M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z\"><\/path><path id=\"MJX-5711-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-5711-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5711-TEX-I-1D450\" d=\"M34 159Q34 268 120 355T306 442Q362 442 394 418T427 355Q427 326 408 306T360 285Q341 285 330 295T319 325T330 359T352 380T366 386H367Q367 388 361 392T340 400T306 404Q276 404 249 390Q228 381 206 359Q162 315 142 235T121 119Q121 73 147 50Q169 26 205 26H209Q321 26 394 111Q403 121 406 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mathvariant=\"bold\">x<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold x<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.109ex\" height=\"1.464ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 490 647\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.464ex;\"><defs><path id=\"MJX-2129-TEX-I-1D466\" d=\"M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 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Examples<\/span><\/h1><h2 id='7-estimator-of-a-matrix-channel'><span>7. 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transform=\"translate(2399.6,0)\"><use data-c=\"1D460\" xlink:href=\"#MJX-5819-TEX-I-1D460\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(3090.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5819-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4091,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5819-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>y<\/mi><mo>=<\/mo><mi>h<\/mi><mi>s<\/mi><mo>+<\/mo><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">y=hs+n<\/script><span> with <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.011ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 889 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-257-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-257-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">K<\/script><span> antenna elements at the transmitter and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.378ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1051 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-258-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D440\" xlink:href=\"#MJX-258-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>M<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">M<\/script><span> at the receiver, which means <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.389ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 1940 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-259-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-259-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-259-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(889,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-259-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>K<\/mi><mi>M<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">KM<\/script><span> transmission channels are estimated.<\/span><\/p><p><span>And we compare three linear estimators<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n747\" cid=\"n747\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 18.957ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.993ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.931ex; min-width: 18.957ex;\"><defs><path id=\"MJX-5712-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><path id=\"MJX-5712-TEX-B-5E\" d=\"M207 632L287 694Q289 693 368 632T448 570T431 545T413 520Q410 520 350 559L287 597L224 559Q164 520 161 520Q160 520 143 544T126 570T207 632Z\"><\/path><path id=\"MJX-5712-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path 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637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -911.5)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 911.5) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3389.4 -911.5 1 1323\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-161.5)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mover\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-5712-TEX-B-1D421\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(319.5,279) translate(-287.5 0)\"><use data-c=\"5E\" xlink:href=\"#MJX-5712-TEX-B-5E\"><\/use><\/g><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(916.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5712-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1972.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D413\" xlink:href=\"#MJX-5712-TEX-B-1D413\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2772.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5712-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3657.3,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-5712-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(4602.1,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"2102\" xlink:href=\"#MJX-5712-TEX-D-2102\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(755,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D43E\" xlink:href=\"#MJX-5712-TEX-I-1D43E\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(889,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5712-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -911.5 1 1323\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-161.5)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mover><mi mathvariant=\"bold\">h<\/mi><mo mathvariant=\"bold\" stretchy=\"false\">^<\/mo><\/mover><\/mrow><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">T<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>\u2208<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>K<\/mi><mi>M<\/mi><\/mrow><\/msup><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>MMSE, mean square error estimator<\/span><\/p><\/li><li><p><span>ML, maximum likelihood estimator<\/span><\/p><\/li><li><p><span>MF, matched filter estimator<\/span><\/p><\/li><\/ul><h3 id='71-channel-model'><span>7.1 Channel Model<\/span><\/h3><h4 id='the-task'><span>The task <\/span><\/h4><p><span>is to find good estimators of the channel coefficients<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n758\" cid=\"n758\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 44.036ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.347ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.608ex; min-width: 44.036ex;\"><defs><path id=\"MJX-5713-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-5713-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5713-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-5713-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><path id=\"MJX-5713-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-5713-TEX-N-77\" d=\"M90 368Q84 378 76 380T40 385H18V431H24L43 430Q62 430 84 429T116 428Q206 428 221 431H229V385H215Q177 383 177 368Q177 367 221 239L265 113L339 328L333 345Q323 374 316 379Q308 384 278 385H258V431H264Q270 428 348 428Q439 428 454 431H461V385H452Q404 385 404 369Q404 366 418 324T449 234T481 143L496 100L537 219Q579 341 579 347Q579 363 564 373T530 385H522V431H529Q541 428 624 428Q692 428 698 431H703V385H697Q696 385 691 385T682 384Q635 377 619 334L559 161Q546 124 528 71Q508 12 503 1T487 -11H479Q460 -11 456 -4Q455 -3 407 133L361 267Q359 263 266 -4Q261 -11 243 -11H238Q225 -11 220 -3L90 368Z\"><\/path><path id=\"MJX-5713-TEX-N-69\" d=\"M69 609Q69 637 87 653T131 669Q154 667 171 652T188 609Q188 579 171 564T129 549Q104 549 87 564T69 609ZM247 0Q232 3 143 3Q132 3 106 3T56 1L34 0H26V46H42Q70 46 91 49Q100 53 102 60T104 102V205V293Q104 345 102 359T88 378Q74 385 41 385H30V408Q30 431 32 431L42 432Q52 433 70 434T106 436Q123 437 142 438T171 441T182 442H185V62Q190 52 197 50T232 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360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5713-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-5713-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><path id=\"MJX-5713-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-5713-TEX-N-61\" d=\"M137 305T115 305T78 320T63 359Q63 394 97 421T218 448Q291 448 336 416T396 340Q401 326 401 309T402 194V124Q402 76 407 58T428 40Q443 40 448 56T453 109V145H493V106Q492 66 490 59Q481 29 455 12T400 -6T353 12T329 54V58L327 55Q325 52 322 49T314 40T302 29T287 17T269 6T247 -2T221 -8T190 -11Q130 -11 82 20T34 107Q34 128 41 147T68 188T116 225T194 253T304 268H318V290Q318 324 312 340Q290 411 215 411Q197 411 181 410T156 406T148 403Q170 388 170 359Q170 334 154 320ZM126 106Q126 75 150 51T209 26Q247 26 276 49T315 109Q317 116 318 175Q318 233 317 233Q309 233 296 232T251 223T193 203T147 166T126 106Z\"><\/path><path id=\"MJX-5713-TEX-N-6E\" d=\"M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 442Q450 438 463 329Q464 322 464 190V104Q464 66 466 59T477 49Q498 46 526 46H542V0H534L510 1Q487 2 460 2T422 3Q319 3 310 0H302V46H318Q379 46 379 62Q380 64 380 200Q379 335 378 343Q372 371 358 385T334 402T308 404Q263 404 229 370Q202 343 195 315T187 232V168V108Q187 78 188 68T191 55T200 49Q221 46 249 46H265V0H257L234 1Q210 2 183 2T145 3Q42 3 33 0H25V46H41Z\"><\/path><path id=\"MJX-5713-TEX-N-64\" d=\"M376 495Q376 511 376 535T377 568Q377 613 367 624T316 637H298V660Q298 683 300 683L310 684Q320 685 339 686T376 688Q393 689 413 690T443 693T454 694H457V390Q457 84 458 81Q461 61 472 55T517 46H535V0Q533 0 459 -5T380 -11H373V44L365 37Q307 -11 235 -11Q158 -11 96 50T34 215Q34 315 97 378T244 442Q319 442 376 393V495ZM373 342Q328 405 260 405Q211 405 173 369Q146 341 139 305T131 211Q131 155 138 120T173 59Q203 26 251 26Q322 26 373 103V342Z\"><\/path><path id=\"MJX-5713-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -768.6)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 768.6) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"8932 -768.6 1 1037.2\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,18.6)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-5713-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5713-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5713-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1156,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-5713-TEX-I-1D458\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtext\" transform=\"translate(1844.8,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5713-TEX-N-A0\"><\/use><use data-c=\"77\" xlink:href=\"#MJX-5713-TEX-N-77\" transform=\"translate(250,0)\"><\/use><use data-c=\"69\" xlink:href=\"#MJX-5713-TEX-N-69\" transform=\"translate(972,0)\"><\/use><use data-c=\"74\" xlink:href=\"#MJX-5713-TEX-N-74\" transform=\"translate(1250,0)\"><\/use><use data-c=\"68\" xlink:href=\"#MJX-5713-TEX-N-68\" transform=\"translate(1639,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-5713-TEX-N-A0\" transform=\"translate(2195,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4289.8,0)\"><use data-c=\"1D45A\" xlink:href=\"#MJX-5713-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5445.6,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5713-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(6501.4,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5713-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7001.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5713-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(7446,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5713-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8784.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5713-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(9229.4,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5713-TEX-I-1D440\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(10280.4,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5713-TEX-N-A0\"><\/use><use data-c=\"61\" xlink:href=\"#MJX-5713-TEX-N-61\" transform=\"translate(250,0)\"><\/use><use data-c=\"6E\" xlink:href=\"#MJX-5713-TEX-N-6E\" transform=\"translate(750,0)\"><\/use><use data-c=\"64\" xlink:href=\"#MJX-5713-TEX-N-64\" transform=\"translate(1306,0)\"><\/use><use data-c=\"A0\" xlink:href=\"#MJX-5713-TEX-N-A0\" transform=\"translate(1862,0)\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12392.4,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-5713-TEX-I-1D458\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13191.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5713-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(14246.9,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5713-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(14746.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5713-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15191.6,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5713-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16530.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5713-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(16974.9,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-5713-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -768.6 1 1037.2\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,18.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>m<\/mi><mo>,<\/mo><mi>k<\/mi><\/mrow><\/msub><mtext>&nbsp;with&nbsp;<\/mtext><mi>m<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>M<\/mi><mtext>&nbsp;and&nbsp;<\/mtext><mi>k<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>K<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.174ex\" height=\"2.22ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1844.8 981.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-265-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-265-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-265-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-265-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-265-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-265-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-265-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1156,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-265-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>m<\/mi><mo>,<\/mo><mi>k<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_{m,k}<\/script><span> denotes the channel coefficient from the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.179ex\" height=\"1.595ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 521 705\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-261-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D458\" xlink:href=\"#MJX-261-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>k<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">k<\/script><span>th transmitter to the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.986ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 878 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-262-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-262-TEX-I-1D45A\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>m<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">m<\/script><span>th receiver.<\/span><\/p><h4 id='the-traning-signals'><span>The traning signals <\/span><\/h4><p><span>consist of <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.009ex\" height=\"1.545ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -683 888 683\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-2145-TEX-I-1D441\" d=\"M234 637Q231 637 226 637Q201 637 196 638T191 649Q191 676 202 682Q204 683 299 683Q376 683 387 683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-2145-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>N<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">N<\/script><span> vectors <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"21.748ex\" height=\"2.353ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -846 9612.7 1040\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.439ex;\"><defs><path id=\"MJX-264-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-264-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-264-TEX-N-2208\" d=\"M84 250Q84 372 166 450T360 539Q361 539 377 539T419 540T469 540H568Q583 532 583 520Q583 511 570 501L466 500Q355 499 329 494Q280 482 242 458T183 409T147 354T129 306T124 272V270H568Q583 262 583 250T568 230H124V228Q124 207 134 177T167 112T231 48T328 7Q355 1 466 0H570Q583 -10 583 -20Q583 -32 568 -40H471Q464 -40 446 -40T417 -41Q262 -41 172 45Q84 127 84 250Z\"><\/path><path id=\"MJX-264-TEX-D-2102\" d=\"M684 131Q684 125 672 109T633 71T573 29T489 -5T386 -19Q330 -19 276 -3T174 46T91 134T44 261Q39 283 39 341T44 421Q66 538 143 611T341 699Q344 699 364 700T395 701Q449 698 503 677T585 655Q603 655 611 662T620 678T625 694T639 702Q650 702 657 690V481L653 474Q640 467 628 472Q624 476 618 496T595 541Q562 587 507 625T390 663H381Q337 663 299 625Q212 547 212 336Q212 249 233 179Q274 30 405 30Q533 30 641 130Q658 147 666 147Q671 147 677 143T684 131ZM250 625Q264 643 261 643Q238 635 214 620T161 579T110 510T79 414Q74 384 74 341T79 268Q89 213 113 169T164 101T217 61T260 39L277 34Q270 41 264 48Q199 111 181 254Q178 281 178 344T181 434Q200 559 250 625ZM621 565V625Q617 623 613 623Q603 619 590 619H575L588 605Q608 583 610 579L621 565Z\"><\/path><path id=\"MJX-264-TEX-I-1D43E\" d=\"M285 628Q285 635 228 637Q205 637 198 638T191 647Q191 649 193 661Q199 681 203 682Q205 683 214 683H219Q260 681 355 681Q389 681 418 681T463 682T483 682Q500 682 500 674Q500 669 497 660Q496 658 496 654T495 648T493 644T490 641T486 639T479 638T470 637T456 637Q416 636 405 634T387 623L306 305Q307 305 490 449T678 597Q692 611 692 620Q692 635 667 637Q651 637 651 648Q651 650 654 662T659 677Q662 682 676 682Q680 682 711 681T791 680Q814 680 839 681T869 682Q889 682 889 672Q889 650 881 642Q878 637 862 637Q787 632 726 586Q710 576 656 534T556 455L509 418L518 396Q527 374 546 329T581 244Q656 67 661 61Q663 59 666 57Q680 47 717 46H738Q744 38 744 37T741 19Q737 6 731 0H720Q680 3 625 3Q503 3 488 0H478Q472 6 472 9T474 27Q478 40 480 43T491 46H494Q544 46 544 71Q544 75 517 141T485 216L427 354L359 301L291 248L268 155Q245 63 245 58Q245 51 253 49T303 46H334Q340 37 340 35Q340 19 333 5Q328 0 317 0Q314 0 280 1T180 2Q118 2 85 2T49 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q147 65 216 339T285 628Z\"><\/path><path id=\"MJX-264-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-264-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-264-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z\"><\/path><path id=\"MJX-264-TEX-N-2026\" d=\"M78 60Q78 84 95 102T138 120Q162 120 180 104T199 61Q199 36 182 18T139 0T96 17T78 60ZM525 60Q525 84 542 102T585 120Q609 120 627 104T646 61Q646 36 629 18T586 0T543 17T525 60ZM972 60Q972 84 989 102T1032 120Q1056 120 1074 104T1093 61Q1093 36 1076 18T1033 0T990 17T972 60Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-264-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(487,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-264-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1239,0)\"><use data-c=\"2208\" xlink:href=\"#MJX-264-TEX-N-2208\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2183.8,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"2102\" xlink:href=\"#MJX-264-TEX-D-2102\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(755,363) scale(0.707)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-264-TEX-I-1D43E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3617.4,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-264-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(4062.1,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-264-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4939.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-264-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5995.7,0)\"><use data-c=\"31\" xlink:href=\"#MJX-264-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6495.7,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-264-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6940.3,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-264-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8279,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-264-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(8723.7,0)\"><use data-c=\"1D43E\" xlink:href=\"#MJX-264-TEX-I-1D43E\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mi>n<\/mi><\/msub><mo>\u2208<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"double-struck\">C<\/mi><\/mrow><mi>K<\/mi><\/msup><mo>,<\/mo><mi>n<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>K<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold s_n\\in\\C^K,n=1,\\dots,K<\/script><span>.<\/span><\/p><h4 id='the-estimation'><span>The estimation <\/span><\/h4><p><span>of the channel coefficients <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.174ex\" height=\"2.22ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 1844.8 981.2\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.65ex;\"><defs><path id=\"MJX-265-TEX-I-210E\" d=\"M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z\"><\/path><path id=\"MJX-265-TEX-I-1D45A\" d=\"M21 287Q22 293 24 303T36 341T56 388T88 425T132 442T175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q303 442 384 442Q401 442 415 440T441 433T460 423T475 411T485 398T493 385T497 373T500 364T502 357L510 367Q573 442 659 442Q713 442 746 415T780 336Q780 285 742 178T704 50Q705 36 709 31T724 26Q752 26 776 56T815 138Q818 149 821 151T837 153Q857 153 857 145Q857 144 853 130Q845 101 831 73T785 17T716 -10Q669 -10 648 17T627 73Q627 92 663 193T700 345Q700 404 656 404H651Q565 404 506 303L499 291L466 157Q433 26 428 16Q415 -11 385 -11Q372 -11 364 -4T353 8T350 18Q350 29 384 161L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 181Q151 335 151 342Q154 357 154 369Q154 405 129 405Q107 405 92 377T69 316T57 280Q55 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-265-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-265-TEX-I-1D458\" d=\"M121 647Q121 657 125 670T137 683Q138 683 209 688T282 694Q294 694 294 686Q294 679 244 477Q194 279 194 272Q213 282 223 291Q247 309 292 354T362 415Q402 442 438 442Q468 442 485 423T503 369Q503 344 496 327T477 302T456 291T438 288Q418 288 406 299T394 328Q394 353 410 369T442 390L458 393Q446 405 434 405H430Q398 402 367 380T294 316T228 255Q230 254 243 252T267 246T293 238T320 224T342 206T359 180T365 147Q365 130 360 106T354 66Q354 26 381 26Q429 26 459 145Q461 153 479 153H483Q499 153 499 144Q499 139 496 130Q455 -11 378 -11Q333 -11 305 15T277 90Q277 108 280 121T283 145Q283 167 269 183T234 206T200 217T182 220H180Q168 178 159 139T145 81T136 44T129 20T122 7T111 -2Q98 -11 83 -11Q66 -11 57 -1T48 16Q48 26 85 176T158 471L195 616Q196 629 188 632T149 637H144Q134 637 131 637T124 640T121 647Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"mi\"><use data-c=\"210E\" xlink:href=\"#MJX-265-TEX-I-210E\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(609,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D45A\" xlink:href=\"#MJX-265-TEX-I-1D45A\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(878,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-265-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(1156,0)\"><use data-c=\"1D458\" xlink:href=\"#MJX-265-TEX-I-1D458\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mi>h<\/mi><mrow data-mjx-texclass=\"ORD\"><mi>m<\/mi><mo>,<\/mo><mi>k<\/mi><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">h_{m,k}<\/script><span> is based on the received signal vectors <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n764\" cid=\"n764\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 32.855ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.262ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex; min-width: 32.855ex;\"><defs><path id=\"MJX-5714-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-5714-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><path id=\"MJX-5714-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5714-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-5714-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-5714-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5714-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-5714-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-5714-TEX-N-A0\" d=\"\"><\/path><path id=\"MJX-5714-TEX-N-31\" d=\"M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 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100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"6461 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5714-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(640,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5714-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1392,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5714-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2447.8,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-5714-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(3347.8,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-5714-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(487,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5714-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4531.3,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5714-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(5531.5,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5714-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(672,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5714-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6677.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5714-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(7122.5,0)\"><use data-c=\"A0\" xlink:href=\"#MJX-5714-TEX-N-A0\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(7372.5,0)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5714-TEX-I-1D45B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8250.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5714-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(9306,0)\"><use data-c=\"31\" xlink:href=\"#MJX-5714-TEX-N-31\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(9806,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5714-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10250.7,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5714-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11589.3,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5714-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(12034,0)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5714-TEX-I-1D441\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mi>n<\/mi><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mi>n<\/mi><\/msub><mo>+<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mi>n<\/mi><\/msub><mo>,<\/mo><mtext>&nbsp;<\/mtext><mi>n<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><mi>N<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.036ex\" height=\"1.552ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -686 900 686\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-266-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-266-TEX-B-1D407\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold H<\/script><span> denotes the matrix of channel coefficients, <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.521ex\" height=\"1.457ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -444 1114.3 644\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-267-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-267-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-267-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(640,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-267-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mi>n<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold y_n<\/script><span> the received signal vector, and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"2.593ex\" height=\"1.375ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -450 1146.3 607.8\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.357ex;\"><defs><path id=\"MJX-268-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-268-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-268-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(672,-150) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-268-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mi>n<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold n_n<\/script><span> the noise corruption at the receiver for the <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.357ex\" height=\"1.025ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -442 600 453\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.025ex;\"><defs><path id=\"MJX-269-TEX-I-1D45B\" d=\"M21 287Q22 293 24 303T36 341T56 388T89 425T135 442Q171 442 195 424T225 390T231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336T465 179T427 52Q427 26 444 26Q450 26 453 27Q482 32 505 65T540 145Q542 153 560 153Q580 153 580 145Q580 144 576 130Q568 101 554 73T508 17T439 -10Q392 -10 371 17T350 73Q350 92 386 193T423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 112 180T152 343Q153 348 153 366Q153 405 129 405Q91 405 66 305Q60 285 60 284Q58 278 41 278H27Q21 284 21 287Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D45B\" xlink:href=\"#MJX-269-TEX-I-1D45B\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>n<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">n<\/script><span>th traning vector.<\/span><\/p><h4 id='the-model'><span>The model <\/span><\/h4><p><span>for the training channel is <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n768\" cid=\"n768\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div 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683T401 677Q612 181 616 168L670 381Q723 592 723 606Q723 633 659 637Q635 637 635 648Q635 650 637 660Q641 676 643 679T653 683Q656 683 684 682T767 680Q817 680 843 681T873 682Q888 682 888 672Q888 650 880 642Q878 637 858 637Q787 633 769 597L620 7Q618 0 599 0Q585 0 582 2Q579 5 453 305L326 604L261 344Q196 88 196 79Q201 46 268 46H278Q284 41 284 38T282 19Q278 6 272 0H259Q228 2 151 2Q123 2 100 2T63 2T46 1Q31 1 31 10Q31 14 34 26T39 40Q41 46 62 46Q130 49 150 85Q154 91 221 362L289 634Q287 635 234 637Z\"><\/path><path id=\"MJX-5715-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><path id=\"MJX-5715-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5715-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-5715-TEX-B-1D42C\" d=\"M38 315Q38 339 45 360T70 404T127 440T223 453Q273 453 320 436L338 445L357 453H366Q380 453 383 447T386 403V387V355Q386 331 383 326T365 321H355H349Q333 321 329 324T324 341Q317 406 224 406H216Q123 406 123 353Q123 334 143 321T188 304T244 294T285 286Q305 281 325 273T373 237T412 172Q414 162 414 142Q414 -6 230 -6Q154 -6 117 22L68 -6H58Q44 -6 41 0T38 42V73Q38 85 38 101T37 122Q37 144 42 148T68 153H75Q87 153 91 151T97 147T103 132Q131 46 220 46H230Q257 46 265 47Q330 58 330 108Q330 127 316 142Q300 156 284 162Q271 168 212 178T122 202Q38 243 38 315Z\"><\/path><path id=\"MJX-5715-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5715-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"9325.2 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mo\"><use data-c=\"5B\" xlink:href=\"#MJX-5715-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(278,0)\"><g data-mml-node=\"TeXAtom\" 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xlink:href=\"#MJX-5715-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4867.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5715-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(5423.2,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5715-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6479,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-5715-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7379,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5715-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(7657,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-5715-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(487,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5715-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8547.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5715-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(8992.2,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5715-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(10330.9,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5715-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(10775.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D42C\" xlink:href=\"#MJX-5715-TEX-B-1D42C\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(487,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5715-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(11940.5,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5715-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(12440.7,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5715-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13440.9,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-5715-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(13718.9,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5715-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"mn\" transform=\"translate(672,-150) scale(0.707)\"><use data-c=\"31\" xlink:href=\"#MJX-5715-TEX-N-31\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(14794.5,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5715-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(15239.1,0)\"><use data-c=\"2026\" xlink:href=\"#MJX-5715-TEX-N-2026\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16577.8,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5715-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(17022.5,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5715-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(672,-150) scale(0.707)\"><use data-c=\"1D441\" xlink:href=\"#MJX-5715-TEX-I-1D441\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(18372.4,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5715-TEX-N-5D\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mi>N<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">s<\/mi><\/mrow><mi>N<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><mo>+<\/mo><mo stretchy=\"false\">[<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mn>1<\/mn><\/msub><mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mi>N<\/mi><\/msub><mo stretchy=\"false\">]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>i.e.,<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block 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327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5716-TEX-B-1D407\" d=\"M400 0Q376 3 226 3Q75 3 51 0H39V62H147V624H39V686H51Q75 683 226 683Q376 683 400 686H412V624H304V388H595V624H487V686H499Q523 683 673 683Q824 683 848 686H860V624H752V62H860V0H848Q824 3 674 3Q523 3 499 0H487V62H595V326H304V62H412V0H400Z\"><\/path><path id=\"MJX-5716-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-5716-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5716-TEX-B-1D40D\" d=\"M314 0Q296 3 181 3T48 0H39V62H147V624H39V686H171H265Q288 686 297 686T309 684T315 679Q317 676 500 455T684 233V624H576V686H585Q603 683 718 683T851 686H860V624H752V319Q752 15 750 11Q747 4 742 2T718 0H712Q708 0 706 0T700 0T696 1T693 2T690 4T687 7T684 11T679 16T674 23Q671 27 437 311L215 579V62H323V0H314Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -750)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 750) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"2932 -750 1 1000\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D418\" xlink:href=\"#MJX-5716-TEX-B-1D418\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1146.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5716-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2202.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D407\" xlink:href=\"#MJX-5716-TEX-B-1D407\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3102.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-5716-TEX-B-1D412\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3963.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5716-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4964,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D40D\" xlink:href=\"#MJX-5716-TEX-B-1D40D\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -750 1 1000\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">Y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">H<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">N<\/mi><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>By stacking the column vectors of the matrices, we obtain<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n772\" cid=\"n772\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 23.772ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"3.709ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -1.289ex; min-width: 23.772ex;\"><defs><path id=\"MJX-5717-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-5717-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5717-TEX-N-21\" d=\"M78 661Q78 682 96 699T138 716T180 700T199 661Q199 654 179 432T158 206Q156 198 139 198Q121 198 119 206Q118 209 98 431T78 661ZM79 61Q79 89 97 105T141 121Q164 119 181 104T198 61Q198 31 181 16T139 1Q114 1 97 16T79 61Z\"><\/path><path id=\"MJX-5717-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5717-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-5717-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-5717-TEX-N-2297\" d=\"M56 250Q56 394 156 488T384 583Q530 583 626 485T722 250Q722 110 625 14T390 -83Q249 -83 153 14T56 250ZM582 471Q531 510 496 523Q446 542 381 542Q324 542 272 519T196 471L389 278L485 375L582 471ZM167 442Q95 362 95 250Q95 137 167 58L359 250L167 442ZM610 58Q682 138 682 250Q682 363 610 442L418 250L610 58ZM196 29Q209 16 230 2T295 -27T388 -42Q409 -42 429 -40T465 -33T496 -23T522 -11T544 1T561 13T574 22T582 29L388 222L196 29Z\"><\/path><path id=\"MJX-5717-TEX-B-1D408\" d=\"M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z\"><\/path><path id=\"MJX-5717-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><path id=\"MJX-5717-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5717-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><path id=\"MJX-5717-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-5717-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -1069.6)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1069.6) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"4453.6 -1069.6 1 1639.3\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-319.6)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-5717-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mover\" transform=\"translate(884.8,0)\"><g data-mml-node=\"mo\"><use data-c=\"3D\" xlink:href=\"#MJX-5717-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(290.7,783) scale(0.707)\"><use data-c=\"21\" xlink:href=\"#MJX-5717-TEX-N-21\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1940.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5717-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2329.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-5717-TEX-B-1D412\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(672,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5717-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3755.3,0)\"><use data-c=\"2297\" xlink:href=\"#MJX-5717-TEX-N-2297\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4755.5,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-5717-TEX-B-1D408\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(469,-150) scale(0.707)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5717-TEX-I-1D440\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(6017.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5717-TEX-N-29\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(6406.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-5717-TEX-B-1D421\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(7267.9,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-5717-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(8268.2,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5717-TEX-B-1D427\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1069.6 1 1639.3\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-319.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mover><mo>=<\/mo><mo>!<\/mo><\/mover><mo stretchy=\"false\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo>\u2297<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">I<\/mi><\/mrow><mi>M<\/mi><\/msub><mo stretchy=\"false\">)<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>! here we use<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n774\" cid=\"n774\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 44.23ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.601ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.735ex; min-width: 44.23ex;\"><defs><path id=\"MJX-5718-TEX-B-1D400\" d=\"M296 0Q278 3 164 3Q58 3 49 0H40V62H92Q144 62 144 64Q388 682 397 689Q403 698 434 698Q463 698 471 689Q475 686 538 530T663 218L724 64Q724 62 776 62H828V0H817Q796 3 658 3Q509 3 485 0H472V62H517Q561 62 561 63L517 175H262L240 120Q218 65 217 64Q217 62 261 62H306V0H296ZM390 237L492 238L440 365Q390 491 388 491Q287 239 287 237H390Z\"><\/path><path id=\"MJX-5718-TEX-B-1D417\" d=\"M327 0Q306 3 174 3Q52 3 43 0H33V62H98L162 63L360 333L157 624H48V686H59Q80 683 217 683Q368 683 395 686H408V624H335L393 540L452 458L573 623Q573 624 528 624H483V686H494Q515 683 646 683Q769 683 778 686H787V624H658L575 511Q493 398 493 397L508 376Q522 356 553 312T611 229L727 62H835V0H824Q803 3 667 3Q516 3 489 0H476V62H513L549 63L401 274L247 63Q247 62 292 62H338V0H327Z\"><\/path><path id=\"MJX-5718-TEX-B-1D401\" d=\"M720 510Q720 476 704 448T665 404T619 377T580 362L564 359L583 356Q602 353 632 342T690 312Q712 292 725 276Q752 235 752 189V183Q752 160 741 125Q698 18 547 2Q543 1 288 0H39V62H147V624H39V686H264H409Q502 686 542 681T624 655Q720 607 720 510ZM563 513Q563 553 548 578T518 611T486 622Q479 624 385 624H293V382H375Q458 383 467 385Q563 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367 744 368L739 374Q734 379 726 390T707 416T685 453T663 499Q658 511 658 515Q658 525 680 525Q687 524 690 523T695 519T701 507Q766 359 902 287Q921 276 939 269T961 259T966 250Q966 246 965 244T960 240T949 236T930 228T902 213Q763 137 701 -7Q697 -16 695 -19T690 -23T680 -25Q658 -25 658 -15Q658 -11 663 1Q673 24 685 46T707 83T725 109T739 126L744 132Q744 133 500 133Q255 133 255 132L260 126Q265 121 273 110T292 84T314 47T336 1Q341 -11 341 -15Q341 -25 319 -25Q312 -24 309 -23T304 -19T298 -7Q233 141 97 213Q83 221 70 227T51 235T41 239T35 243T34 250T35 256T40 261T51 265T70 273T97 287Q235 363 299 509Q305 522 308 524ZM792 319L783 327H216Q183 294 120 256L110 250L120 244Q173 212 207 181L216 173H783L792 181Q826 212 879 244L889 250L879 256Q826 288 792 319Z\"><\/path><path id=\"MJX-5718-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 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431H508V385H497Q439 381 423 345Q421 341 356 172T288 -2Q283 -11 263 -11Q244 -11 239 -2Q99 359 98 364Q93 378 82 381T43 385H19V431H25L33 430Q41 430 53 430T79 430T104 429T122 428Q217 428 232 431H240V385H226Q187 384 184 370Q184 366 235 234L286 102L377 341V349Q377 363 367 372T349 383T335 385H331V431H338Z\"><\/path><path id=\"MJX-5718-TEX-N-65\" d=\"M28 218Q28 273 48 318T98 391T163 433T229 448Q282 448 320 430T378 380T406 316T415 245Q415 238 408 231H126V216Q126 68 226 36Q246 30 270 30Q312 30 342 62Q359 79 369 104L379 128Q382 131 395 131H398Q415 131 415 121Q415 117 412 108Q393 53 349 21T250 -11Q155 -11 92 58T28 218ZM333 275Q322 403 238 411H236Q228 411 220 410T195 402T166 381T143 340T127 274V267H333V275Z\"><\/path><path id=\"MJX-5718-TEX-N-63\" d=\"M370 305T349 305T313 320T297 358Q297 381 312 396Q317 401 317 402T307 404Q281 408 258 408Q209 408 178 376Q131 329 131 219Q131 137 162 90Q203 29 272 29Q313 29 338 55T374 117Q376 125 379 127T395 129H409Q415 123 415 120Q415 116 411 104T395 71T366 33T318 2T249 -11Q163 -11 99 53T34 214Q34 318 99 383T250 448T370 421T404 357Q404 334 387 320Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -824.7)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 824.7) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"8974.8 -824.7 1 1149.5\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\" transform=\"translate(0,-74.7)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-5718-TEX-B-1D400\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(869,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5718-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1738,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D401\" xlink:href=\"#MJX-5718-TEX-B-1D401\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2833.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5718-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3889.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-5718-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4998.3,0)\"><use data-c=\"21D4\" xlink:href=\"#MJX-5718-TEX-N-21D4\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(6276.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5718-TEX-N-28\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(6665.1,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D401\" xlink:href=\"#MJX-5718-TEX-B-1D401\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(851,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-5718-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(8269.9,0)\"><use data-c=\"2297\" xlink:href=\"#MJX-5718-TEX-N-2297\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(9270.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D400\" xlink:href=\"#MJX-5718-TEX-B-1D400\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(10139.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5718-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(10528.1,0)\"><use data-c=\"76\" xlink:href=\"#MJX-5718-TEX-N-76\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-5718-TEX-N-65\" transform=\"translate(528,0)\"><\/use><use data-c=\"63\" xlink:href=\"#MJX-5718-TEX-N-63\" transform=\"translate(972,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(11944.1,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5718-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(12333.1,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D417\" xlink:href=\"#MJX-5718-TEX-B-1D417\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(13202.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5718-TEX-N-29\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(13868.9,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5718-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mtext\" transform=\"translate(14924.7,0)\"><use data-c=\"76\" xlink:href=\"#MJX-5718-TEX-N-76\"><\/use><use data-c=\"65\" xlink:href=\"#MJX-5718-TEX-N-65\" transform=\"translate(528,0)\"><\/use><use data-c=\"63\" xlink:href=\"#MJX-5718-TEX-N-63\" transform=\"translate(972,0)\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(16340.7,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5718-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(16729.7,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-5718-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(17560.7,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5718-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -824.7 1 1149.5\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-74.7)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">A<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">B<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mo stretchy=\"false\">\u21d4<\/mo><mo stretchy=\"false\">(<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">B<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo>\u2297<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">A<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mtext>vec<\/mtext><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">X<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><mo>=<\/mo><mtext>vec<\/mtext><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mo stretchy=\"false\">)<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='further-assumptions'><span>Further Assumptions<\/span><\/h4><p><span>We further assume that the stacked vector are Gaussian distributed<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n777\" cid=\"n777\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 17.455ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.833ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.351ex; min-width: 17.455ex;\"><defs><path id=\"MJX-5719-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><path id=\"MJX-5719-TEX-N-223C\" d=\"M55 166Q55 241 101 304T222 367Q260 367 296 349T362 304T421 252T484 208T554 189Q616 189 655 236T694 338Q694 350 698 358T708 367Q722 367 722 334Q722 260 677 197T562 134H554Q517 134 481 152T414 196T355 248T292 293T223 311Q179 311 145 286Q109 257 96 218T80 156T69 133Q55 133 55 166Z\"><\/path><path id=\"MJX-5719-TEX-C-4E\" d=\"M343 705Q358 705 358 698Q360 696 370 658T411 524T484 319Q536 174 590 82L595 73L615 152Q646 274 683 407Q729 571 752 637T799 727Q852 780 937 788Q939 788 947 788T958 789H962Q979 789 979 765Q979 722 951 692Q942 683 924 683Q888 681 859 672T818 654T803 639Q784 608 708 322T631 15Q631 14 630 15Q630 17 629 15Q628 14 628 12Q621 -4 601 -17T560 -31Q550 -31 546 -28T530 -7Q484 67 458 123T398 272Q352 392 314 514L306 535V534Q306 533 296 488T272 379T234 239T185 100T127 -7T61 -50Q34 -50 4 -34T-27 8Q-27 33 -12 61T18 90Q21 90 36 77T87 57H92Q109 57 123 78T162 173Q206 299 232 417T265 599T276 667Q284 681 304 693T343 705Z\"><\/path><path id=\"MJX-5719-TEX-N-28\" d=\"M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z\"><\/path><path id=\"MJX-5719-TEX-B-1D7CE\" d=\"M266 654H280H282Q500 654 524 418Q529 370 529 320Q529 125 456 52Q397 -10 287 -10Q110 -10 63 154Q45 212 45 316Q45 504 113 585Q140 618 185 636T266 654ZM374 548Q347 604 286 604Q247 604 218 575Q197 552 193 511T188 311Q188 159 196 116Q202 87 225 64T287 41Q339 41 367 87Q379 107 382 152T386 329Q386 518 374 548Z\"><\/path><path id=\"MJX-5719-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-5719-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-5719-TEX-N-29\" d=\"M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z\"><\/path><path id=\"MJX-5719-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(0.019042,-0.019042) translate(0, -1539)\"><g data-mml-node=\"math\"><g data-mml-node=\"mtable\" transform=\"translate(800,0) translate(-800,0)\"><g transform=\"translate(0 1539) matrix(1 0 0 -1 0 0) scale(52.51)\"><svg data-table=\"true\" preserveAspectRatio=\"xMidYMid\" viewBox=\"3057.5 -1539 1 2578\"><g transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mlabeledtr\"><g data-mml-node=\"mtd\"><g data-mml-node=\"mtable\"><g data-mml-node=\"mtr\" transform=\"translate(0,750)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-5719-TEX-B-1D421\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(916.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-5719-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1972.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-5719-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2951.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5719-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3340.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-5719-TEX-B-1D7CE\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3915.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5719-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4360.2,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-5719-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-5719-TEX-B-1D421\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5726.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5719-TEX-N-29\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mtr\" transform=\"translate(0,-789)\"><g data-mml-node=\"mtd\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5719-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(916.8,0)\"><use data-c=\"223C\" xlink:href=\"#MJX-5719-TEX-N-223C\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1972.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"4E\" xlink:href=\"#MJX-5719-TEX-C-4E\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(2951.6,0)\"><use data-c=\"28\" xlink:href=\"#MJX-5719-TEX-N-28\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(3340.6,0)\"><g data-mml-node=\"mn\"><use data-c=\"1D7CE\" xlink:href=\"#MJX-5719-TEX-B-1D7CE\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3915.6,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-5719-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(4360.2,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-5719-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5719-TEX-B-1D427\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(5726.1,0)\"><use data-c=\"29\" xlink:href=\"#MJX-5719-TEX-N-29\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -1539 1 2578\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtable rowspacing=\".5em\" columnspacing=\"1em\" displaystyle=\"true\"><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><mtr><mtd><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mo>\u223c<\/mo><mrow data-mjx-texclass=\"ORD\"><mi data-mjx-variant=\"-tex-calligraphic\" mathvariant=\"script\">N<\/mi><\/mrow><mo stretchy=\"false\">(<\/mo><mrow data-mjx-texclass=\"ORD\"><mn mathvariant=\"bold\">0<\/mn><\/mrow><mo>,<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/msub><mo stretchy=\"false\">)<\/mo><\/mtd><\/mtr><\/mtable><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>where <\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n779\" cid=\"n779\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 16.443ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"2.576ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.723ex; min-width: 16.443ex;\"><defs><path id=\"MJX-5720-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-5720-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><path id=\"MJX-5720-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-5720-TEX-I-1D70E\" d=\"M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 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xlink:href=\"#MJX-5720-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"msubsup\" transform=\"translate(2699.4,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D70E\" xlink:href=\"#MJX-5720-TEX-I-1D70E\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(604,413) scale(0.707)\"><use data-c=\"32\" xlink:href=\"#MJX-5720-TEX-N-32\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(604,-247) scale(0.707)\"><use data-c=\"1D45B\" xlink:href=\"#MJX-5720-TEX-I-1D45B\"><\/use><\/g><\/g><g data-mml-node=\"msub\" transform=\"translate(3777.7,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-5720-TEX-B-1D408\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(469,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D441\" xlink:href=\"#MJX-5720-TEX-I-1D441\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(888,0)\"><use data-c=\"1D440\" xlink:href=\"#MJX-5720-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -819.4 1 1138.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-64.6)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/msub><mo>=<\/mo><msubsup><mi>\u03c3<\/mi><mi>n<\/mi><mn>2<\/mn><\/msubsup><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">I<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi>N<\/mi><mi>M<\/mi><\/mrow><\/msub><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><p><span>And the channel vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.446ex\" height=\"1.57ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -694 639 694\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-270-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-270-TEX-B-1D421\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold h<\/script><span> and the noise vector <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.446ex\" height=\"1.018ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -450 639 450\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: 0px;\"><defs><path id=\"MJX-271-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-271-TEX-B-1D427\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold n<\/script><span> are assumed to be stochastically independent. and thus uncorrelated.<\/span><\/p><div 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xlink:href=\"#MJX-5721-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3111.7,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-5721-TEX-B-1D427\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(672,413) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5A7\" xlink:href=\"#MJX-5721-TEX-SS-1D5A7\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(4334.3,0)\"><use data-c=\"5D\" xlink:href=\"#MJX-5721-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(4890.1,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-5721-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mn\" transform=\"translate(5945.9,0)\"><use data-c=\"30\" xlink:href=\"#MJX-5721-TEX-N-30\"><\/use><\/g><\/g><\/g><\/g><\/svg><svg data-labels=\"true\" preserveAspectRatio=\"xMaxYMid\" viewBox=\"0 -826.9 1 1153.7\"><g data-labels=\"true\" transform=\"matrix(1 0 0 -1 0 0)\"><g data-mml-node=\"mtd\" id=\"mjx-eqn:\" transform=\"translate(0,-76.9)\"><text data-id-align=\"true\"><\/text><g data-idbox=\"true\" transform=\"translate(0,0)\"><g data-mml-node=\"mrow\"><\/g><\/g><\/g><\/g><\/svg><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"block\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable displaystyle=\"true\"><mlabeledtr><mtd><mrow><\/mrow><\/mtd><mtd><mtext>Cov<\/mtext><mo stretchy=\"false\">[<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><mo>,<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">H<\/mi><\/mrow><\/msup><mo stretchy=\"false\">]<\/mo><mo>=<\/mo><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><h4 id='matrix-\\ns'><span>Matrix <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"1.459ex\" height=\"1.645ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -705 645 727\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.05ex;\"><defs><path id=\"MJX-272-TEX-I-1D446\" d=\"M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 704T632 705Q645 705 645 698T617 577T585 459T569 456Q549 456 549 465Q549 471 550 475Q550 478 551 494T553 520Q553 554 544 579T526 616T501 641Q465 662 419 662Q362 662 313 616T263 510Q263 480 278 458T319 427Q323 425 389 408T456 390Q490 379 522 342T554 242Q554 216 546 186Q541 164 528 137T492 78T426 18T332 -20Q320 -22 298 -22Q199 -22 144 33L134 44L106 13Q83 -14 78 -18T65 -22Q52 -22 52 -14Q52 -11 110 221Q112 227 130 227H143Q149 221 149 216Q149 214 148 207T144 186T142 153Q144 114 160 87T203 47T255 29T308 24Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"mi\"><use data-c=\"1D446\" xlink:href=\"#MJX-272-TEX-I-1D446\"><\/use><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mi>S<\/mi><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">S<\/script><\/h4><p><span>is used as an abbreviation for <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"8.344ex\" height=\"2.261ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 3688.2 999.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.339ex;\"><defs><path id=\"MJX-273-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-273-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-273-TEX-N-2297\" d=\"M56 250Q56 394 156 488T384 583Q530 583 626 485T722 250Q722 110 625 14T390 -83Q249 -83 153 14T56 250ZM582 471Q531 510 496 523Q446 542 381 542Q324 542 272 519T196 471L389 278L485 375L582 471ZM167 442Q95 362 95 250Q95 137 167 58L359 250L167 442ZM610 58Q682 138 682 250Q682 363 610 442L418 250L610 58ZM196 29Q209 16 230 2T295 -27T388 -42Q409 -42 429 -40T465 -33T496 -23T522 -11T544 1T561 13T574 22T582 29L388 222L196 29Z\"><\/path><path id=\"MJX-273-TEX-B-1D408\" d=\"M397 0Q370 3 218 3Q65 3 38 0H25V62H139V624H25V686H38Q65 683 218 683Q370 683 397 686H410V624H296V62H410V0H397Z\"><\/path><path id=\"MJX-273-TEX-I-1D440\" d=\"M289 629Q289 635 232 637Q208 637 201 638T194 648Q194 649 196 659Q197 662 198 666T199 671T201 676T203 679T207 681T212 683T220 683T232 684Q238 684 262 684T307 683Q386 683 398 683T414 678Q415 674 451 396L487 117L510 154Q534 190 574 254T662 394Q837 673 839 675Q840 676 842 678T846 681L852 683H948Q965 683 988 683T1017 684Q1051 684 1051 673Q1051 668 1048 656T1045 643Q1041 637 1008 637Q968 636 957 634T939 623Q936 618 867 340T797 59Q797 55 798 54T805 50T822 48T855 46H886Q892 37 892 35Q892 19 885 5Q880 0 869 0Q864 0 828 1T736 2Q675 2 644 2T609 1Q592 1 592 11Q592 13 594 25Q598 41 602 43T625 46Q652 46 685 49Q699 52 704 61Q706 65 742 207T813 490T848 631L654 322Q458 10 453 5Q451 4 449 3Q444 0 433 0Q418 0 415 7Q413 11 374 317L335 624L267 354Q200 88 200 79Q206 46 272 46H282Q288 41 289 37T286 19Q282 3 278 1Q274 0 267 0Q265 0 255 0T221 1T157 2Q127 2 95 1T58 0Q43 0 39 2T35 11Q35 13 38 25T43 40Q45 46 65 46Q135 46 154 86Q158 92 223 354T289 629Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msup\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-273-TEX-B-1D412\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(672,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-273-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(1425.8,0)\"><use data-c=\"2297\" xlink:href=\"#MJX-273-TEX-N-2297\"><\/use><\/g><g data-mml-node=\"msub\" transform=\"translate(2426,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D408\" xlink:href=\"#MJX-273-TEX-B-1D408\"><\/use><\/g><\/g><g data-mml-node=\"mi\" transform=\"translate(469,-150) scale(0.707)\"><use data-c=\"1D440\" xlink:href=\"#MJX-273-TEX-I-1D440\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo>\u2297<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">I<\/mi><\/mrow><mi>M<\/mi><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold S^\\mathsf{T}\\otimes\\bold I_M<\/script><span>.<\/span><\/p><p><span>Consequently, due to the linear channel model <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"11.493ex\" height=\"2.029ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 5080 897\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.452ex;\"><defs><path id=\"MJX-274-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-274-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-274-TEX-B-1D412\" d=\"M64 493Q64 582 120 636T264 696H272Q280 697 285 697Q380 697 454 645L480 669Q484 672 488 676T495 683T500 688T504 691T508 693T511 695T514 696T517 697T522 697Q536 697 539 691T542 652V577Q542 557 542 532T543 500Q543 472 540 465T524 458H511H505Q489 458 485 461T479 478Q472 529 449 564T393 614T336 634T287 639Q228 639 203 610T177 544Q177 517 195 493T247 457Q253 454 343 436T475 391Q574 326 574 207V200Q574 163 559 120Q517 12 389 -9Q380 -10 346 -10Q308 -10 275 -5T221 7T184 22T160 35T151 40L126 17Q122 14 118 10T111 3T106 -2T102 -5T98 -7T95 -9T92 -10T89 -11T84 -11Q70 -11 67 -4T64 35V108Q64 128 64 153T63 185Q63 203 63 211T69 223T77 227T94 228H100Q118 228 122 225T126 205Q130 125 193 88T345 51Q408 51 434 82T460 157Q460 196 439 221T388 257Q384 259 305 276T221 295Q155 313 110 366T64 493Z\"><\/path><path id=\"MJX-274-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><path id=\"MJX-274-TEX-N-2B\" d=\"M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z\"><\/path><path id=\"MJX-274-TEX-B-1D427\" d=\"M40 442Q217 450 218 450H224V407L225 365Q233 378 245 391T289 422T362 448Q374 450 398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V210V303Q106 353 104 363T91 376Q77 380 50 380H37V442H40Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-274-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(884.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-274-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(1940.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D412\" xlink:href=\"#MJX-274-TEX-B-1D412\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(2579.6,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-274-TEX-B-1D421\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3440.8,0)\"><use data-c=\"2B\" xlink:href=\"#MJX-274-TEX-N-2B\"><\/use><\/g><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\" transform=\"translate(4441,0)\"><g data-mml-node=\"mi\"><use data-c=\"1D427\" xlink:href=\"#MJX-274-TEX-B-1D427\"><\/use><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><mo>+<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold y=\\bold S\\bold h+\\bold n<\/script><span>, we conclude <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"13.088ex\" height=\"2.488ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -849.5 5784.8 1099.5\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.566ex;\"><defs><path id=\"MJX-275-TEX-B-1D433\" d=\"M48 262Q48 264 54 349T60 436V444H252Q289 444 336 444T394 445Q441 445 450 441T459 418Q459 406 458 404Q456 399 327 229T194 55H237Q260 56 268 56T297 58T325 65T348 77T370 98T384 128T395 170Q400 197 400 216Q400 217 431 217H462V211Q461 208 453 108T444 6V0H245Q46 0 43 2Q32 7 32 28V33Q32 41 40 52T84 112Q129 170 164 217L298 393H256Q189 392 165 380Q124 360 115 303Q110 280 110 256Q110 254 79 254H48V262Z\"><\/path><path id=\"MJX-275-TEX-N-3D\" d=\"M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z\"><\/path><path id=\"MJX-275-TEX-N-5B\" d=\"M118 -250V750H255V710H158V-210H255V-250H118Z\"><\/path><path id=\"MJX-275-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><path id=\"MJX-275-TEX-SS-1D5B3\" d=\"M36 608V688H644V608H518L392 609V0H288V609L162 608H36Z\"><\/path><path id=\"MJX-275-TEX-N-2C\" d=\"M78 35T78 60T94 103T137 121Q165 121 187 96T210 8Q210 -27 201 -60T180 -117T154 -158T130 -185T117 -194Q113 -194 104 -185T95 -172Q95 -168 106 -156T131 -126T157 -76T173 -3V9L172 8Q170 7 167 6T161 3T152 1T140 0Q113 0 96 17Z\"><\/path><path id=\"MJX-275-TEX-B-1D421\" d=\"M40 686L131 690Q222 694 223 694H229V533L230 372L238 381Q248 394 264 407T317 435T398 450Q428 450 448 447T491 434T529 402T551 346Q553 335 554 198V62H623V0H614Q596 3 489 3Q374 3 365 0H356V62H425V194V275Q425 348 416 373T371 399Q326 399 288 370T238 290Q236 281 235 171V62H304V0H295Q277 3 171 3Q64 3 46 0H37V62H106V332Q106 387 106 453T107 534Q107 593 105 605T91 620Q77 624 50 624H37V686H40Z\"><\/path><path id=\"MJX-275-TEX-N-5D\" d=\"M22 710V750H159V-250H22V-210H119V710H22Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D433\" xlink:href=\"#MJX-275-TEX-B-1D433\"><\/use><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(788.8,0)\"><use data-c=\"3D\" xlink:href=\"#MJX-275-TEX-N-3D\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(1844.6,0)\"><use data-c=\"5B\" xlink:href=\"#MJX-275-TEX-N-5B\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(2122.6,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-275-TEX-B-1D432\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(640,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-275-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"mo\" transform=\"translate(3294.1,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-275-TEX-N-2C\"><\/use><\/g><g data-mml-node=\"msup\" transform=\"translate(3738.8,0)\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D421\" xlink:href=\"#MJX-275-TEX-B-1D421\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(672,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-275-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><g data-mml-node=\"msup\" transform=\"translate(4942.3,0)\"><g data-mml-node=\"mo\"><use data-c=\"5D\" xlink:href=\"#MJX-275-TEX-N-5D\"><\/use><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(311,363) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D5B3\" xlink:href=\"#MJX-275-TEX-SS-1D5B3\"><\/use><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">z<\/mi><\/mrow><mo>=<\/mo><mo stretchy=\"false\">[<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><mo>,<\/mo><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><msup><mo stretchy=\"false\">]<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">T<\/mi><\/mrow><\/msup><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold z=[\\bold y^\\mathsf{T},\\bold h^\\mathsf{T}]^\\mathsf{T}<\/script><span> are jointly Gaussian distributed. The covariance is equal to<\/span><\/p><div contenteditable=\"false\" spellcheck=\"false\" class=\"mathjax-block md-end-block md-math-block md-rawblock\" id=\"mathjax-n785\" cid=\"n785\" mdtype=\"math_block\" data-math-tag-before=\"2\" data-math-tag-after=\"2\" data-math-labels=\"[]\"><div class=\"md-rawblock-container md-math-container\" tabindex=\"-1\"><mjx-container class=\"MathJax\" jax=\"SVG\" display=\"true\" width=\"full\" style=\"min-width: 23.317ex; position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"100%\" height=\"5.617ex\" role=\"img\" focusable=\"false\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -2.243ex; min-width: 23.317ex;\"><defs><path id=\"MJX-5722-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 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columnspacing=\"0em\" rowspacing=\"3pt\"><mtr><mtd><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/msub><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">H<\/mi><\/mrow><\/msup><mo>+<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">n<\/mi><\/mrow><\/msub><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><mo mathvariant=\"bold\">,<\/mo><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/mrow><\/msub><mo>=<\/mo><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mi>h<\/mi><\/msub><\/mtd><\/mtr><mtr><mtd><\/mtd><mtd><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><mo mathvariant=\"bold\">,<\/mo><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/mrow><\/msub><mo>=<\/mo><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/msub><msup><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">S<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"sans-serif\">H<\/mi><\/mrow><\/msup><\/mtd><\/mtr><\/mtable><\/math><\/mjx-assistive-mml><\/mjx-container><\/div><\/div><ul><li><p><span>In the following, we assume full knowledge of the covariance matrices <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"3.039ex\" height=\"2.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 1343.2 988.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.659ex;\"><defs><path id=\"MJX-276-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 639T657 632L694 663Q703 670 711 677T723 687T730 692T735 695T740 696T746 697Q759 697 762 692T766 668V627V489V449Q766 428 762 424T742 419H732H720Q699 419 697 436Q690 498 657 545Q611 618 532 632Q522 634 496 634Q356 634 286 553Q232 488 232 343T286 133Q355 52 497 52Q597 52 650 112T704 237Q704 248 709 251T729 254H735Q750 254 755 253T763 248T766 234Q766 136 680 63T469 -11Q285 -11 175 86T64 343Z\"><\/path><path id=\"MJX-276-TEX-B-1D432\" d=\"M84 -102Q84 -110 87 -119T102 -138T133 -149Q148 -148 162 -143T186 -131T206 -114T222 -95T234 -76T243 -59T249 -45T252 -37L269 0L96 382H26V444H34Q49 441 146 441Q252 441 270 444H279V382H255Q232 382 232 380L337 151L442 382H394V444H401Q413 441 495 441Q568 441 574 444H580V382H510L406 152Q298 -84 297 -87Q269 -139 225 -169T131 -200Q85 -200 54 -172T23 -100Q23 -64 44 -50T87 -35Q111 -35 130 -50T152 -92V-100H84V-102Z\"><\/path><\/defs><g stroke=\"currentColor\" fill=\"currentColor\" stroke-width=\"0\" transform=\"scale(1,-1)\"><g data-mml-node=\"math\"><g data-mml-node=\"msub\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D402\" xlink:href=\"#MJX-276-TEX-B-1D402\"><\/use><\/g><\/g><g data-mml-node=\"TeXAtom\" transform=\"translate(864,-150) scale(0.707)\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-276-TEX-B-1D432\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><\/mrow><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold C_{\\bold {y}}<\/script><span> and <\/span><mjx-container class=\"MathJax\" jax=\"SVG\" style=\"position: relative;\"><svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"4.572ex\" height=\"2.236ex\" role=\"img\" focusable=\"false\" viewBox=\"0 -697 2020.6 988.4\" xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" aria-hidden=\"true\" style=\"vertical-align: -0.659ex;\"><defs><path id=\"MJX-277-TEX-B-1D402\" d=\"M64 343Q64 502 174 599T468 697Q502 697 533 691T586 674T623 655T647 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data-mjx-texclass=\"ORD\"><g data-mml-node=\"TeXAtom\" data-mjx-texclass=\"ORD\"><g data-mml-node=\"mi\"><use data-c=\"1D432\" xlink:href=\"#MJX-277-TEX-B-1D432\"><\/use><\/g><g data-mml-node=\"mo\" transform=\"translate(607,0)\"><use data-c=\"2C\" xlink:href=\"#MJX-277-TEX-B-2C\"><\/use><\/g><g data-mml-node=\"mi\" transform=\"translate(926,0)\"><use data-c=\"1D421\" xlink:href=\"#MJX-277-TEX-B-1D421\"><\/use><\/g><\/g><\/g><\/g><\/g><\/g><\/svg><mjx-assistive-mml unselectable=\"on\" display=\"inline\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><msub><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">C<\/mi><\/mrow><mrow data-mjx-texclass=\"ORD\"><mrow data-mjx-texclass=\"ORD\"><mi mathvariant=\"bold\">y<\/mi><mo mathvariant=\"bold\">,<\/mo><mi mathvariant=\"bold\">h<\/mi><\/mrow><\/mrow><\/msub><\/math><\/mjx-assistive-mml><\/mjx-container><script type=\"math\/tex\">\\bold C_{\\bold {y,h}}<\/script><span>.<\/span><\/p><\/li><\/ul><h2 id='72-three-linear-estimator'><span>7.2 Three Linear Estimator<\/span><\/h2><h3 id='1-mmse-minimum-mean-square-error-estimation'><span>1. MMSE (minimum mean square error estimation)<\/span><\/h3><p>&nbsp;<\/p><\/div><\/div>\n<\/body>\n<\/html>\n","protected":false},"excerpt":{"rendered":"<p>Statistical Signal Processing I. Parameter Estimation 1<a class=\"more-link\" href=\"https:\/\/blog.mhrooz.xyz\/index.php\/2023\/06\/17\/statistical_signal_processing_-_summary\/\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">&#8220;Statistical Signal Processing &#8211; Summary&#8221;<\/span><\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[49],"tags":[72],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2815"}],"collection":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/comments?post=2815"}],"version-history":[{"count":1,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2815\/revisions"}],"predecessor-version":[{"id":2816,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/posts\/2815\/revisions\/2816"}],"wp:attachment":[{"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/media?parent=2815"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/categories?post=2815"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.mhrooz.xyz\/index.php\/wp-json\/wp\/v2\/tags?post=2815"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}