Polyhedral Star-Shaped Distributions

<div><svg height="5.97156pt" id="M511" style="vertical-align:-0.04980993pt" version="1.1" viewbox="-0.0498162 -5.92175 9.16851 5.97156" width="9.16851pt" 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></svg>-stars in case <svg height="8.98582pt" id="M512" style="vertical-align:-0.6370001pt" version="1.1" viewbox="-0.0498162 -8.34882 51.542 8.98582" width="51.542pt" 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="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,-0.013,9.872,0)"><path d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z" id="g117-91"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,21.135,0)"><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,-0.013,33.802,0)"><path d="M512 -3V55L134 254V256L512 456V514L75 281V230L512 -3Z" id="g117-91"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,45.065,0)"><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></svg> (a) and in case <svg height="9.46863pt" id="M513" style="vertical-align:-1.11981pt" version="1.1" viewbox="-0.0498162 -8.34882 30.3552 9.46863" width="30.3552pt" 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="M531 285V335L57 550V494L473 310L57 127V71L531 285ZM531 -40V10H57V-40H531Z" id="g117-94"></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="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></svg> (b).</div>

Journal of Probability and Statistics

Polyhedral Star-Shaped Distributions

Figure 1