Type-2 Gumbel distribution

In probability theory, the Type-2 Gumbel probability density function is

${\displaystyle f(x|a,b)=abx^{-a-1}e^{-bx^{-a}}\,}$

Type-2 Gumbel

Parameters	${\displaystyle a\!}$ (real) ${\displaystyle b\!}$ shape (real)
PDF	${\displaystyle abx^{-a-1}e^{-bx^{-a}}\!}$
CDF	${\displaystyle e^{-bx^{-a}}\!}$
Mean	${\displaystyle b^{1/a}\Gamma (1-1/a)\!}$
Variance	${\displaystyle b^{2/a}(\Gamma (1-1/a)-{\Gamma (1-1/a)}^{2})\!}$

for

${\displaystyle 0<x<\infty }$.

This implies that it is similar to the Weibull distributions, substituting ${\displaystyle b=\lambda ^{-k}}$ and ${\displaystyle a=-k}$. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a, which is not allowed here as it would yield a negative probability density.

For ${\displaystyle 0<a\leq 1}$ the mean is infinite. For ${\displaystyle 0<a\leq 2}$ the variance is infinite.

The cumulative distribution function is

${\displaystyle F(x|a,b)=e^{-bx^{-a}}\,}$

The moments ${\displaystyle E[X^{k}]\,}$ exist for ${\displaystyle k<a\,}$

The special case b = 1 yields the Fréchet distribution.

Based on The GNU Scientific Library, used under GFDL.