Type-2 Gumbel distribution

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

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

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

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

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

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

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

Generating random variates

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

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

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

Related distributions

• The special case ${\displaystyle \ b=1\ }$ yields the Fréchet distribution.

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

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

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Based on "Gumbel distribution". The GNU Scientific Library. type 002d2, used under GFDL.

See also

• Extreme value theory
• Gumbel distribution