广义极值分布¶
带一个形状参数的极值分布 \(c\) 。
如果 \(c>0\) ，支持是 \(-\infty<x\leq1/c.\) 如果 \(c<0\) ，支持是 \(\frac{{1}}{{c}}\leq x<\infty.\)
\Begin{eqnarray*}f\Left(x；c\Right)&=&\exp\left(-\left(1-cx\right)^{1/c}\right)\left(1-cx\right)^{1/c-1}\\
F\Left(x；c\Right)&=&\exp\Left(-\Left(1-cx\Right)^{1/c}\Right)\\
g\Left(q；c\Right)&=&\frac{1}{c}\Left(1-\Left(-\logq\Right)^{c}\Right)\end{eqnarray*}
\[\mu{n}^{\Prime}=\frac{1}{c^{n}}\sum_{k=0}^{n}\binom{n}{k}\left(-1\right)^{k}\Gamma\left(ck+1\right)\quad\text{if}cn>-1\]
所以,
\BEGIN{eqnarray*}\mu{1}^{\Prime}&=&\frac{1}{c}\left(1-\Gamma\left(1+c\right)\right)\quad c>-1\\
\MU_{2}^{\PRIME}&=&\frac{1}{c^{2}}\left(1-2\Gamma\left(1+c\right)+\Gamma\left(1+2c\right)\right)\quad c>-\FRAC{1}{2}\\
\MU_{3}^{\PRIME}&=&\frac{1}{c^{3}}\left(1-3\Gamma\left(1+c\right)+3\Gamma\left(1+2c\right)-\Gamma\left(1+3c\right)\right)\quad c>-\FRAC{1}{3}\\
\MU_{4}^{\PRIME}&=&\frac{1}{c^{4}}\left(1-4\Gamma\left(1+c\right)+6\Gamma\left(1+2c\right)-4\Gamma\left(1+3c\right)+\Gamma\left(1+4c\right)\right)\quad c>-\FRAC{1}{4}\END{等式*}
为 \(c=0\) 该分布与(左偏的)Gumbel分布相同，支撑为 \(\mathbb{{R}}\) 。
\BEGIN{eqnarray*}f\Left(x；0\Right)&=&\exp\Left(-e^{-x}\Right)e^{-x}\\
F\Left(x；0\Right)&=&\exp\Left(-e^{-x}\Right)\\
g\Left(Q；0\Right)&=&-\log\Left(-\log Q\Right)\end{eqnarray*}
\BEGIN{eqnarray*}\mu&=&\Gamma=-\psi_{0}\Left(1\Right)\\
\MU_{2}&=&\frac{\pi^{2}}{6}\\
\Gamma_{1}&=&\frac{12\sqrt{6}}{\pi^{3}}\zeta\left(3\right)\\
\Gamma_{2}&=&\frac{12}{5}\end{eqnarray*}