<svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 12.6536 11.6718" width="12.6536pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-86" d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z"/></g></svg>-Statistic for Multivariate Stable Distributions

<div>RMSEs of <svg height="12.6786pt" id="M224" style="vertical-align:-3.42939pt" version="1.1" viewbox="-0.0498162 -9.24921 6.63704 12.6786" width="6.63704pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,1.05,-.222)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" id="g113-225"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="501" vert-adv-y="501"></glyph.data></g></svg> under different scenarios when <svg height="8.55521pt" id="M225" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 44.1999 8.55521" width="44.1999pt" 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="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z" id="g113-223"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="568" vert-adv-y="568"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.017,0)"><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,-0.013,22.28,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,28.52,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,31.484,0)"><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><g transform="matrix(.013,0,0,-0.013,37.724,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z" id="g113-54"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>. We use the following symbol scheme: ⋄ for <svg height="13.1856pt" id="M226" style="vertical-align:-3.832001pt" version="1.1" viewbox="-0.0498162 -9.3536 22.3134 13.1856" width="22.3134pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.32,-.326)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M498 369C498 423 469 454 440 454C415 454 388 432 388 408C388 398 391 390 395 384C401 375 413 354 413 330C413 269 332 132 280 70H278C285 133 286 231 273 319C259 412 231 454 191 454C147 454 85 416 21 322L49 293C91 339 110 351 119 351C129 351 132 349 142 324C181 225 194 56 178 -19C148 -80 115 -167 107 -224L125 -246C167 -243 241 -219 265 -202C260 -169 255 -95 256 -21C261 -11 268 2 323 70C438 212 498 291 498 369Z" id="g5-225"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.772,3.668)"><path d="M727 650H480V617C552 612 575 600 580 563C583 531 587 491 587 398V326C587 153 534 44 398 44C281 44 210 124 210 296V519C210 607 218 612 298 617V650H33V617C114 609 123 607 123 519V274C123 153 151 81 211 34C258 -1 314 -14 374 -14C441 -14 509 7 556 56C615 116 633 202 633 322V398C633 491 635 533 639 561C643 600 667 614 727 617V650Z" id="g58-83"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="747" vert-adv-y="747"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,13.451,3.668)"><path d="M871 0V33C783 41 780 45 777 152L768 506C765 610 771 609 860 617V650H683L453 157L228 650H59V617C150 609 154 606 146 481L132 275C124 170 117 125 112 93C105 50 84 39 27 34V0H267V34C200 39 176 48 173 93C170 130 171 172 176 256L190 540H192L414 8H438L682 554H685L689 152C689 45 686 41 604 33V0H871Z" id="g58-75"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="902" vert-adv-y="902"></glyph.data></g></svg>, ○ for <svg height="13.0714pt" id="M227" style="vertical-align:-3.7178pt" version="1.1" viewbox="-0.0498162 -9.3536 23.848 13.0714" width="23.848pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.32,-.326)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M498 369C498 423 469 454 440 454C415 454 388 432 388 408C388 398 391 390 395 384C401 375 413 354 413 330C413 269 332 132 280 70H278C285 133 286 231 273 319C259 412 231 454 191 454C147 454 85 416 21 322L49 293C91 339 110 351 119 351C129 351 132 349 142 324C181 225 194 56 178 -19C148 -80 115 -167 107 -224L125 -246C167 -243 241 -219 265 -202C260 -169 255 -95 256 -21C261 -11 268 2 323 70C438 212 498 291 498 369Z" id="g5-225"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.772,3.668)"><path d="M871 0V33C783 41 780 45 777 152L768 506C765 610 771 609 860 617V650H683L453 157L228 650H59V617C150 609 154 606 146 481L132 275C124 170 117 125 112 93C105 50 84 39 27 34V0H267V34C200 39 176 48 173 93C170 130 171 172 176 256L190 540H192L414 8H438L682 554H685L689 152C689 45 686 41 604 33V0H871Z" id="g58-75"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="902" vert-adv-y="902"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,14.98,3.668)"><path d="M871 0V33C783 41 780 45 777 152L768 506C765 610 771 609 860 617V650H683L453 157L228 650H59V617C150 609 154 606 146 481L132 275C124 170 117 125 112 93C105 50 84 39 27 34V0H267V34C200 39 176 48 173 93C170 130 171 172 176 256L190 540H192L414 8H438L682 554H685L689 152C689 45 686 41 604 33V0H871Z" id="g58-75"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="902" vert-adv-y="902"></glyph.data></g></svg>, + for <svg height="13.1856pt" id="M228" style="vertical-align:-3.832001pt" version="1.1" viewbox="-0.0498162 -9.3536 31.3927 13.1856" width="31.3927pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.32,-.326)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M498 369C498 423 469 454 440 454C415 454 388 432 388 408C388 398 391 390 395 384C401 375 413 354 413 330C413 269 332 132 280 70H278C285 133 286 231 273 319C259 412 231 454 191 454C147 454 85 416 21 322L49 293C91 339 110 351 119 351C129 351 132 349 142 324C181 225 194 56 178 -19C148 -80 115 -167 107 -224L125 -246C167 -243 241 -219 265 -202C260 -169 255 -95 256 -21C261 -11 268 2 323 70C438 212 498 291 498 369Z" id="g5-225"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.772,3.668)"><path d="M871 0V33C783 41 780 45 777 152L768 506C765 610 771 609 860 617V650H683L453 157L228 650H59V617C150 609 154 606 146 481L132 275C124 170 117 125 112 93C105 50 84 39 27 34V0H267V34C200 39 176 48 173 93C170 130 171 172 176 256L190 540H192L414 8H438L682 554H685L689 152C689 45 686 41 604 33V0H871Z" id="g58-75"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="902" vert-adv-y="902"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,14.98,3.668)"><path d="M499 166C484 119 466 88 447 68C424 43 390 38 338 38C297 38 262 40 242 51C223 62 218 80 218 133V522C218 607 226 612 309 617V650H41V617C124 612 131 605 131 522V129C131 45 123 38 38 33V0H497C505 30 529 128 534 160L499 166Z" id="g58-74"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="548" vert-adv-y="548"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,19.967,3.668)"><path d="M300 253H71L57 198H286L300 253Z" id="g190-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="356" vert-adv-y="356"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,23.219,3.668)"><path d="M396 114C349 71 317 59 274 59C216 59 123 104 123 241C123 342 180 400 246 400C283 400 317 386 348 359C355 352 361 348 366 348C378 348 401 371 402 394C402 405 398 413 385 424C369 437 337 451 294 451H293C254 451 193 434 141 394C73 342 39 277 39 198C39 90 114 -12 243 -12C303 -12 371 31 415 90L396 114Z" id="g58-97"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="432" vert-adv-y="432"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,27.15,3.668)"><path d="M56 439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z" id="g58-100"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="304" vert-adv-y="304"></glyph.data></g></svg>, <svg height="7.35473pt" id="M229" style="vertical-align:-0.3499303pt" version="1.1" viewbox="-0.0498162 -7.0048 7.75925 7.35473" width="7.75925pt" 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="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z" id="g117-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g></svg> for <svg height="14.5877pt" id="M230" style="vertical-align:-5.2341pt" version="1.1" viewbox="-0.0498162 -9.3536 29.4745 14.5877" width="29.4745pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.32,-.326)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M498 369C498 423 469 454 440 454C415 454 388 432 388 408C388 398 391 390 395 384C401 375 413 354 413 330C413 269 332 132 280 70H278C285 133 286 231 273 319C259 412 231 454 191 454C147 454 85 416 21 322L49 293C91 339 110 351 119 351C129 351 132 349 142 324C181 225 194 56 178 -19C148 -80 115 -167 107 -224L125 -246C167 -243 241 -219 265 -202C260 -169 255 -95 256 -21C261 -11 268 2 323 70C438 212 498 291 498 369Z" id="g5-225"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.772,3.668)"><path d="M416 501L399 642C356 655 314 665 271 665C140 665 63 582 63 486C63 372 163 325 229 290C304 250 359 217 359 142C359 70 316 25 240 25C138 25 94 125 78 186L44 178C52 123 64 45 72 21C81 16 96 8 121 0S179 -15 220 -15C353 -15 445 68 445 176C445 289 350 335 270 375C191 414 143 449 143 520C143 572 177 627 253 627C340 627 366 556 382 495L416 501Z" id="g58-81"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="484" vert-adv-y="484"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.177,3.668)"><path d="M729 -138C637 -95 547 -41 484 2C600 37 712 151 712 329C712 532 560 665 390 665C173 665 46 499 46 313C46 210 91 125 155 69C217 10 303 -11 357 -14C482 -102 560 -138 592 -146C623 -155 672 -162 720 -162L729 -138ZM351 29C239 53 151 172 151 340C151 531 256 624 365 624C501 624 607 507 607 303C607 142 529 55 447 31L418 51L351 29Z" id="g58-79"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="756" vert-adv-y="756"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,18.056,3.668)"><path d="M300 253H71L57 198H286L300 253Z" id="g190-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="356" vert-adv-y="356"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,21.308,3.668)"><path d="M396 114C349 71 317 59 274 59C216 59 123 104 123 241C123 342 180 400 246 400C283 400 317 386 348 359C355 352 361 348 366 348C378 348 401 371 402 394C402 405 398 413 385 424C369 437 337 451 294 451H293C254 451 193 434 141 394C73 342 39 277 39 198C39 90 114 -12 243 -12C303 -12 371 31 415 90L396 114Z" id="g58-97"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="432" vert-adv-y="432"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,25.239,3.668)"><path d="M56 439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z" id="g58-100"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="304" vert-adv-y="304"></glyph.data></g></svg>, and ∆ for <svg height="13.1856pt" id="M231" style="vertical-align:-3.832001pt" version="1.1" viewbox="-0.0498162 -9.3536 29.2279 13.1856" width="29.2279pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,.32,-.326)"><path d="M470 557C393 591 314 624 246 666H224C156 624 76 591 0 557L12 532C86 550 163 573 235 596C309 572 384 550 458 532L470 557Z" id="g119-17"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="470" vert-adv-y="470"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M498 369C498 423 469 454 440 454C415 454 388 432 388 408C388 398 391 390 395 384C401 375 413 354 413 330C413 269 332 132 280 70H278C285 133 286 231 273 319C259 412 231 454 191 454C147 454 85 416 21 322L49 293C91 339 110 351 119 351C129 351 132 349 142 324C181 225 194 56 178 -19C148 -80 115 -167 107 -224L125 -246C167 -243 241 -219 265 -202C260 -169 255 -95 256 -21C261 -11 268 2 323 70C438 212 498 291 498 369Z" id="g5-225"></path><glyph.data ascent="3481" descent="-2878" horiz-adv-x="519" vert-adv-y="519"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,6.772,3.668)"><path d="M618 181C565 77 513 26 413 26C264 26 151 150 151 336C151 486 240 625 407 625C516 625 573 582 600 475L634 483C627 540 622 581 613 638C585 643 518 665 435 665C211 665 46 526 46 315C46 158 157 -15 406 -15C480 -15 564 4 594 12C612 43 638 121 651 168L618 181Z" id="g58-65"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="673" vert-adv-y="673"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,12.897,3.668)"><path d="M502 500C497 551 492 613 491 650H43V617C122 612 130 606 130 521V129C130 45 122 38 41 33V0H319V33C227 38 217 45 217 129V305H319C411 305 413 295 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439L27 407L32 396H103V106C103 41 96 35 31 30V0H271V29C191 37 184 40 184 110V396H293C304 404 309 428 303 439H184V473C183 558 195 604 207 623S234 653 261 653C293 653 322 635 341 616C350 607 360 606 370 614C381 622 386 631 389 638C394 652 393 664 384 675C366 696 339 708 303 710C263 705 225 686 190 655C137 608 121 561 114 538S103 487 103 456V439H56Z" id="g58-100"></path><glyph.data ascent="1007" descent="-369" horiz-adv-x="304" vert-adv-y="304"></glyph.data></g></svg>.</div>

Journal of Probability and Statistics

fig3

Figure 3

Figure 3: <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 12.6536 11.6718" width="12.6536pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-86" d="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z"/></g></svg>-Statistic for Multivariate Stable Distributions 