Computing the Moments of Order Statistics from Independent Nonidentically Distributed Exponentiated Frechet Variables

<table>Graph of p.d.f. of (inid) o.s.  from Ef distribution for selected values of <svg height="15.7175" id="M30" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 62.285 15.7175" width="62.285" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.7175)">
<text transform="matrix(1,0,0,-1,0.498,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
</text>
<text transform="matrix(1,0,0,-1,7.05,0.498)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
</text>
<text transform="matrix(1,0,0,-1,15.412,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.028002" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279562" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405342" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.656902" y="0">3</tspan>
</text>
</g>
</svg>; <svg height="15.905" id="M31" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 62.285 15.905" width="62.285" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.905)">
<text transform="matrix(1,0,0,-1,0.498,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
</text>
<text transform="matrix(1,0,0,-1,7.05,0.498)">
<tspan style="font-size: 8.75px; " x="0" y="0">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,15.412,3.624)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.028002" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279562" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405342" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.656902" y="0">5</tspan>
</text>
</g>
</svg>; <svg height="15.87125" id="M32" style="vertical-align:-3.25793pt" version="1.1" viewbox="0 0 62.285 15.87125" width="62.285" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,15.87125)">
<text transform="matrix(1,0,0,-1,0.498,3.746)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
</text>
<text transform="matrix(1,0,0,-1,7.05,0.621)">
<tspan style="font-size: 8.75px; " x="0" y="0">3</tspan>
</text>
<text transform="matrix(1,0,0,-1,15.412,3.746)">
<tspan style="font-size: 12.50px; " x="0" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="12.028002" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="18.279562" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="21.405342" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.656902" y="0">7</tspan>
</text>
</g>
</svg>; <svg height="11.9825" id="M33" style="vertical-align:-0.13794pt" version="1.1" viewbox="0 0 36.841251 11.9825" width="36.841251" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.9825)">
<text transform="matrix(1,0,0,-1,0.498,0.636)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜎</tspan>
<tspan style="font-size: 12.50px; " x="10.17754" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.218044" y="0">1</tspan>
</text>
</g>
</svg>, <svg height="12.02875" id="M34" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 36.216251 12.02875" width="36.216251" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,12.02875)">
<text transform="matrix(1,0,0,-1,0.498,0.673)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑛</tspan>
<tspan style="font-size: 12.50px; " x="9.6774149" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.71792" y="0">3</tspan>
</text>
</g>
</svg>, <svg height="14.01375" id="M35" style="vertical-align:-1.76814pt" version="1.1" viewbox="0 0 63.477501 14.01375" width="63.477501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.01375)">
<text transform="matrix(1,0,0,-1,0.498,2.261)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑟</tspan>
<tspan style="font-size: 12.50px; " x="8.5646372" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="20.605143" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="26.856701" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="32.070503" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="38.322063" y="0">,</tspan>
<tspan style="font-size: 12.50px; " x="43.523361" y="0">3</tspan>
</text>
</g>
</svg>, <i>respectively</i>.</table>

Journal of Probability and Statistics

fig1

Figure 1

Figure 1: Computing the Moments of Order Statistics from Independent Nonidentically Distributed Exponentiated Frechet Variables 