威布尔最大极值分布#
定义为 \(x<0\) 和 \(c>0\) 。
\begin{eqnarray*} f\left(x;c\right) & = & c\left(-x\right)^{c-1}\exp\left(-\left(-x\right)^{c}\right)\\ F\left(x;c\right) & = & \exp\left(-\left(-x\right)^{c}\right)\\ G\left(q;c\right) & = & -\left(-\log q\right)^{1/c}\end{eqnarray*}
均值是上面给出的右偏 Fréchet 分布的负值，其他统计参数可以从以下公式计算
\[\mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).\]
\begin{eqnarray*} \mu & = & -\Gamma\left(1+\frac{1}{c}\right) \\ \mu_{2} & = & \Gamma\left(1+\frac{2}{c}\right) - \Gamma^{2}\left(1+\frac{1}{c}\right) \\ \gamma_{1} & = & -\frac{\Gamma\left(1+\frac{3}{c}\right) - 3\Gamma\left(1+\frac{2}{c}\right)\Gamma\left(1+\frac{1}{c}\right) + 2\Gamma^{3}\left(1+\frac{1}{c}\right)} {\mu_{2}^{3/2}} \\ \gamma_{2} & = & \frac{\Gamma\left(1+\frac{4}{c}\right) - 4\Gamma\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{3}{c}\right) + 6\Gamma^{2}\left(1+\frac{1}{c}\right)\Gamma\left(1+\frac{2}{c}\right) - 3\Gamma^{4}\left(1+\frac{1}{c}\right)} {\mu_{2}^{2}} - 3 \\ m_{d} & = & \begin{cases} -\left(\frac{c-1}{c}\right)^{\frac{1}{c}} & \text{if}\; c > 1 \\ 0 & \text{if}\; c <= 1 \end{cases} \\ m_{n} & = & -\ln\left(2\right)^{\frac{1}{c}} \end{eqnarray*}
\[h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
其中 \(\gamma\) 是欧拉常数，等于
\[\gamma\approx0.57721566490153286061.\]