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Weibull Minimum Extreme Value Distribution¶
A type of extreme-value distribution with a lower bound. Defined for \(x>0\) and \(c>0\)
\begin{eqnarray*}
f\left(x;c\right) & = & cx^{c-1}\exp\left(-x^{c}\right) \\
F\left(x;c\right) & = & 1 - \exp\left(-x^{c}\right) \\
G\left(q;c\right) & = & \left[-\log\left(1-q\right)\right]^{1/c}
\end{eqnarray*}
\[\mu_{n}^{\prime}=\Gamma\left(1+\frac{n}{c}\right)\]
\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*}
\[h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]
where \(\gamma\) is Euler’s constant and equal to
\[\gamma\approx0.57721566490153286061.\]
Implementation: scipy.stats.weibull_min