Polyhedral Star-Shaped Distributions

<div><svg height="14.4911pt" id="M579" style="vertical-align:-5.52898pt" version="1.1" viewbox="-0.0498162 -8.96212 24.9294 14.4911" width="24.9294pt" 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="M542 242C542 369 469 429 379 468C370 472 372 496 376 513L416 681L400 691L344 648L219 51C167 69 113 128 113 217C113 324 166 387 266 428L252 459C124 418 23 336 23 202C23 95 90 13 205 -8L182 -132C165 -222 191 -254 209 -261L215 -258L270 -7C415 -4 542 98 542 242ZM469 232C469 126 403 33 279 39L355 409C399 394 469 331 469 232Z" id="g113-243"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,7.02,3.132)"><path d="M554 433L549 437C539 437 513 441 503 443C481 447 456 451 436 451C354 451 268 415 216 368C150 308 99 205 99 106C99 24 136 -12 165 -12C196 -12 243 14 268 33C318 70 374 118 415 181H419L395 39C374 -84 332 -149 295 -175C269 -193 245 -197 212 -197C137 -197 93 -156 76 -102C71 -87 61 -87 51 -96C36 -110 24 -130 24 -147C24 -183 76 -257 171 -257C226 -257 285 -227 323 -200C387 -155 455 -76 480 94C499 225 538 391 554 433ZM460 386C457 361 438 287 426 254C408 218 373 176 332 133C294 93 246 61 217 61C198 61 186 80 186 122C186 167 206 250 231 294C262 349 288 375 317 387C334 394 358 400 381 400C413 400 439 393 460 386Z" id="g50-104"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="578" vert-adv-y="578"></glyph.data></g><g transform="matrix(.0065,0,0,-0.0065,12.052,5.233)"><path d="M455 312C455 472 400 635 248 635C95 635 39 458 39 312C39 144 93 -12 248 -12C400 -12 455 166 455 312ZM361 314C361 173 337 30 249 30C163 30 131 174 131 313C131 451 155 594 248 594C334 594 361 454 361 314Z" id="g176-49"></path><glyph.data ascent="3439" descent="-2858" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,15.774,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z" id="g50-45"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="237" vert-adv-y="237"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,17.93,3.132)"><path d="M695 450H651L553 110H549L473 450H344L253 111H250L174 450H20L126 0H269L350 306H353L426 0H567L695 450ZM521 44H460L376 406H437L521 44ZM222 44H161L76 406H138L222 44Z" id="g197-60"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="676" vert-adv-y="676"></glyph.data></g></svg> with <svg height="8.55521pt" id="M580" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 45.8571 8.55521" width="45.8571pt" 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="M695 450H651L553 110H549L473 450H344L253 111H250L174 450H20L126 0H269L350 306H353L426 0H567L695 450ZM521 44H460L376 406H437L521 44ZM222 44H161L76 406H138L222 44Z" id="g197-60"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="676" vert-adv-y="676"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,12.668,0)"><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,-0.013,23.931,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,30.171,0)"><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,-0.013,33.135,0)"><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,-0.013,39.375,0)"><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> (a) and its level sets (b).</div>

Journal of Probability and Statistics

Polyhedral Star-Shaped Distributions

Figure 2