Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution

<div>(a) Numerical reconstruction of <svg height="17.7728pt" id="M387" style="vertical-align:-6.1814pt" version="1.1" viewbox="-0.0498162 -11.5914 37.5074 17.7728" width="37.5074pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z" id="g198-5"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="885" vert-adv-y="885"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.505,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.003,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="483" vert-adv-y="483"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.285,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,27.845,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,21.437,3.784)"><path d="M525 275C525 381 464 451 380 451C338 451 289 439 241 405C158 347 127 277 106 152C51 -168 37 -216 24 -242L28 -257C42 -256 92 -247 117 -222C126 -213 134 -193 150 -69L166 32C184 10 211 -12 245 -12C276 -12 332 8 384 46C458 100 525 176 525 275ZM436 260C436 131 366 41 267 41C246 41 208 51 175 82C181 125 187 162 195 200C220 322 257 400 328 400C398 400 436 338 436 260Z" id="g50-239"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="549" vert-adv-y="549"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,32.812,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> in <svg height="8.55521pt" id="M388" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 27.1713 8.55521" width="27.1713pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z" id="g185-40"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="447" vert-adv-y="447"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.495,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.758,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z" id="g113-51"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> dimensions. (b) Numerical reconstruction of <svg height="17.7728pt" id="M389" style="vertical-align:-6.1814pt" version="1.1" viewbox="-0.0498162 -11.5914 37.5074 17.7728" width="37.5074pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z" id="g198-5"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="885" vert-adv-y="885"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.505,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.003,0)"><path d="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="483" vert-adv-y="483"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,22.285,-5.741)"><path d="M556 236V289H56V236H556Z" id="g54-33"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="611" vert-adv-y="611"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,27.845,-5.741)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z" id="g50-50"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="487" vert-adv-y="487"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,21.437,3.784)"><path d="M525 275C525 381 464 451 380 451C338 451 289 439 241 405C158 347 127 277 106 152C51 -168 37 -216 24 -242L28 -257C42 -256 92 -247 117 -222C126 -213 134 -193 150 -69L166 32C184 10 211 -12 245 -12C276 -12 332 8 384 46C458 100 525 176 525 275ZM436 260C436 131 366 41 267 41C246 41 208 51 175 82C181 125 187 162 195 200C220 322 257 400 328 400C398 400 436 338 436 260Z" id="g50-239"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="549" vert-adv-y="549"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,32.812,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> in <svg height="8.55521pt" id="M390" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 27.1713 8.55521" width="27.1713pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z" id="g185-40"></path><glyph.data ascent="989" descent="-303" horiz-adv-x="447" vert-adv-y="447"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.495,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.758,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg> dimensions.</div>

Journal of Probability and Statistics

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Figure 2

Figure 2: Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution 