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