Fitting the Distribution of Linear Combinations of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619999pt" id="M1" height="10.7762pt" version="1.1" viewBox="-0.0657574 -10.3142 16.1157 10.7762" width="16.1157pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g><g transform="matrix(.017,0,0,-0.017,5.852,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg> Variables with more than 2 Degrees of Freedom

<div>PDF of the (normalized) linear combination of <svg height="8.68572pt" id="M246" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 9.95144 8.68572" width="9.95144pt" 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="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"></path></g></svg> Student’s <i>t</i> RVs. We set <svg height="12.5794pt" id="M247" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 40.0658 12.5794" width="40.0658pt" 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="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,6.76,3.132)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.448,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.71,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.26,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.621,0)"><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"></path></g></svg> and <span class="nowrap"><svg height="12.5794pt" id="M248" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 58.9385 12.5794" width="58.9385pt" 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="M213 67C239 136 257 225 255 300C253 353 242 447 164 447C112 447 54 389 41 319L67 309C90 362 117 377 134 377C184 377 189 277 190 239C193 153 164 43 146 -12H176C278 73 450 253 450 390C450 439 410 463 396 463C384 463 380 459 382 449C385 440 389 415 389 402C389 284 312 146 216 67H213Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.902,3.132)"><path d="M250 606C250 634 233 656 203 656C168 656 146 618 146 593C146 564 169 545 192 545C227 545 250 573 250 606ZM227 95L212 119C187 98 152 71 135 71C129 71 128 78 134 102L207 373C219 418 217 451 194 451C165 451 92 411 30 351L44 326C77 353 106 371 114 371C124 371 121 357 117 341L55 97C32 5 46 -12 70 -12C108 -12 191 51 227 95Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.59,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.852,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,32.998,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.534,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.013,0,0,-0.013,47.084,0)"><path d="M368 703H309L44 -163H104L368 703Z"></path></g><g transform="matrix(.013,0,0,-0.013,52.445,0)"><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"></path></g></svg>.</span> Straight lines correspond to the empirical PDF obtained through the Monte Carlo simulations, while dotted lines correspond to our proposed Student’s <i>t</i> distribution fitting based on the matching of the second moments and the CFs for <svg height="6.1673pt" id="M249" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 5.60619 6.1673" width="5.60619pt" 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="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"></path></g></svg> given in (<a href="https://static.hindawi.com/articles/jps/volume-2023/9967290/figures/#EEq40">40</a>).</div>

Journal of Probability and Statistics

fig8

Figure 8

Figure 8: Fitting the Distribution of Linear Combinations of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4619999pt" id="M1" height="10.7762pt" version="1.1" viewBox="-0.0657574 -10.3142 16.1157 10.7762" width="16.1157pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-117" d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"/></g><g transform="matrix(.017,0,0,-0.017,5.852,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g></svg> Variables with more than 2 Degrees of Freedom 

Journal of Probability and Statistics

Fitting the Distribution of Linear Combinations of Variables with more than 2 Degrees of Freedom

Figure 8