Fréchet (left-skewed, Extreme Value Type III, Weibull maximum) Distribution

Defined for \(x<0\) and \(c>0\) .

\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*}

The mean is the negative of the right-skewed Frechet distribution given above, and the other statistical parameters can be computed from

\[\mu_{n}^{\prime}=\left(-1\right)^{n}\Gamma\left(1+\frac{n}{c}\right).\]

\[h\left[X\right]=-\frac{\gamma}{c}-\log\left(c\right)+\gamma+1\]

where \(\gamma\) is Euler’s constant and equal to

\[\gamma\approx0.57721566490153286061.\]

Implementation: scipy.stats.frechet_l