广义帕累托分布¶
有一个形状参数 \(c\neq0\) 。支持是 \(x\geq0\) 如果 \(c>0\) ，以及 \(0\leq x<\frac{{1}}{{\left|c\right|}}\) 如果 \(c\) 是阴性的。
\begin{eqnarray*} f\left(x;c\right) & = & \left(1+cx\right)^{-1-\frac{1}{c}}\\
F\left(x;c\right) & = & 1-\frac{1}{\left(1+cx\right)^{1/c}}\\
G\left(q;c\right) & = & \frac{1}{c}\left[\left(\frac{1}{1-q}\right)^{c}-1\right]\end{eqnarray*}
\[\begin{split}M\left(t\right) = \left\{
\begin{array}{cc}
\left(-\frac{t}{c}\right)^{\frac{1}{c}}
e^{-\frac{t}{c}}
\left[
\Gamma\left(1-\frac{1}{c}\right)
+ \left(\gamma\left(-\frac{1}{c},-\frac{t}{c}\right) / \Gamma\left(\frac{1}{-c}\right)\right)
- \pi\csc\left(\frac{\pi}{c}\right)/\Gamma\left(\frac{1}{c}\right)
\right] & c>0\\
\left(
\frac{\left|c\right|}{t}\right)^{1/\left|c\right|}
\Gamma\left(\frac{1}{\left|c\right|}, \frac{t}{\left|c\right|}\right)
\frac{1}{\Gamma\left(\frac{1}{|c|}\right)}
& c<0
\end{array}
\right.\end{split}\]
\[\mu_{n}^{\prime}=\frac{\left(-1\right)^{n}}{c^{n}}\sum_{k=0}^{n}\binom{n}{k}\frac{\left(-1\right)^{k}}{1-ck}\quad\Text{if}cn<1\]
\BEGIN{eqnarray*}\mu{1}^{\Prime}&=&\frac{1}{1-c}\quad c<1\\
\MU_{2}^{\PRIME}&=&\frac{2}{\left(1-2c\right)\left(1-c\right)}\quad c<\FRAC{1}{2}\\
\MU_{3}^{\PRIME}&=&\frac{6}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)}\quad c<\FRAC{1}{3}\\
\MU_{4}^{\PRIME}&=&\frac{24}{\left(1-c\right)\left(1-2c\right)\left(1-3c\right)\left(1-4c\right)}\quad c<\FRAC{1}{4}\END{等式*}
因此，
\BEGIN{eqnarray*}\MU&=&\MU_{1}^{\PRIME}\\
\MU_{2}&=&\MU_{2}^{\PRIME}-\MU^{2}\\
\Gamma_{1}&=&\frac{\mu_{3}^{\prime}-3\mu\mu_{2}-\mu^{3}}{\mu_{2}^{3/2}}\\
\Gamma_{2}&=&\frac{\mu_{4}^{\prime}-4\mu\mu_{3}-6\mu^{2}\mu_{2}-\mu^{4}}{\mu_{2}^{2}}-3\end{eqnarray*}
\[H\Left [X\right] =1+c\quad c>0\]