Type-2 Gumbel distribution

Short description: Probability distribution

Type-2 Gumbel

Parameters	$a\in \mathbb{ℝ}$ (shape), $b\in \mathbb{ℝ}$ (scale)
Support	$0<x<\mathrm{\infty }$
PDF	$ab{x}^{-a-1}{e}^{-b{x}^{-a}}$
CDF	$e^{-b{x}^{-a}}$
Quantile	${(-\frac{\ln \left(p\right)}{b})}^{-\frac{1}{a}}$
Mean	$b^{\frac{1}{a}}\mathrm{\Gamma }(1-\frac{1}{a})$
Median	${\left(\frac{\ln \left(2\right)}{b}\right)}^{-\frac{1}{a}}$
Mode	${\left(\frac{ab}{a+1}\right)}^{\frac{1}{a}}$
Variance	$b^{\frac{2}{a}}\mathrm{\Gamma }(1-\frac{1}{a})(1-\mathrm{\Gamma }(1-\frac{1}{a}))$
Entropy	$\gamma (1+\frac{1}{a})+\ln \left(\frac{b^{\frac{1}{a}}}{a}\right)+1$

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

$f(x|a,b)=ab{x}^{-a-1}{e}^{-b{x}^{-a}}$ for $x>0.$

For $0<a\le 1$ the mean is infinite. For $0<a\le 2$ the variance is infinite.

The cumulative distribution function is

$F(x|a,b)=e^{-b{x}^{-a}}.$

The moments $\mathbb{𝔼}\left[X^k\right]$ exist for $k<a$

The distribution is named after Emil Julius Gumbel (1891 – 1966).

Generating random variates

Given a random variate $U$ drawn from the uniform distribution in the interval $(0,1),$ then the variate

$X={(-\frac{\ln U}{b})}^{-\frac{1}{a}}$

has a Type-2 Gumbel distribution with parameter $a$ and $b.$ This is obtained by applying the inverse transform sampling-method.

Related distributions

• The special case $b=1$ yields the Fréchet distribution.

• Substituting $b={\lambda }^{-k}$ and $a=-k$ yields the Weibull distribution. Note, however, that a positive $k$ (as in the Weibull distribution) would yield a negative $a$ and hence a negative probability density, which is not allowed.

• If $X$ is Type-2 Gumbel-distributed with parameters $a$ and $b$, then $X^{-1}\sim \mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}(a,b^{-1/a})$.

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Based on "Gumbel distribution". https://www.gnu.org/software/gsl/manual/html_node/The-Type_002d2-Gumbel-Distribution.html,  used under GFDL.

See also

• Extreme value theory
• Gumbel distribution