Home page for accesible maths 11 Other useful distributions 11.4 The log Normal Distribution 11.6 Links Between Standard Distributions

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11.5 The Extreme Value (aka Gumbel) Distribution

Parameters : $\boldsymbol{\theta}=(\alpha,\beta)$ with a location parameter $\alpha\in\mathbb{R}$ and a scale parameter $\beta>0$ .

$f_{X}(x;\boldsymbol{\theta})=\frac{1}{\beta}\exp\{-(x-\alpha)/\beta\}\exp[-% \exp\{-(x-\alpha)/\beta\}]$ for $-\infty<x<\infty$ ,
$F_{X}(x)=\exp[-\exp\{-(x-\alpha)/\beta\}]$ ,
$\operatorname{\mathsf{E}}\left[{X}\right]=\alpha+\beta\gamma$ , where $\gamma\approx 0.5772$ is Euler’s constant,
${\operatorname{\mathsf{Var}}}\left[{X}\right]=\frac{\beta^{2}\pi^{2}}{6}$ .

We write $X\sim\operatorname{\mathsf{GEV}}(\alpha,\beta,0)$ .

Transformations: If $X$ has a $\operatorname{Weibull}(\alpha,\beta)$ distribution, then $-\log(X)$ has an extreme value distribution with location parameter $\log(\beta)$ and scale parameter $1/\alpha$ . In particular, if $X$ is $\operatorname{Exp}(\lambda)$ -distributed, i.e. $X$ is $\operatorname{Weibull}(1,\lambda)$ -distributed, then $-\log(X)$ has an extreme value distribution with location parameter $\log(\lambda)$ and scale parameter $1$ .

Usage: Used to model extreme events such as the height of the biggest wave in a day or the highest or lowest temperature or rainfall amount in a month.

Example 11.5.1.

Let $X_{1}$ and $X_{2}$ be iid with $\operatorname{\mathsf{Gumbel}}(\alpha,\beta)$ distributions. Show that $Y=\max(X_{1},X_{2})$ also has a Gumbel distribution and find its parameters.

Solution. First note that $Y\leq y$ if and only if $X_{1}\leq y$ and $X_{2}\leq y$ , so

	$\displaystyle F_{Y}(y)$	$\displaystyle={\color[rgb]{0.76,0.01,0}F_{X_{1}}(y)F_{X_{2}}(y)}={\color[rgb]{% 0.76,0.01,0}\exp[-\exp\{-(x-\alpha)/\beta\}]\times\exp[-\exp\{-(x-\alpha)/% \beta\}]}$
		$\displaystyle={\color[rgb]{0.76,0.01,0}\exp[-2\exp\{-(x-\alpha)/\beta\}]}$
		$\displaystyle={\color[rgb]{0.76,0.01,0}\exp[-\exp\{-(x-\alpha)/\beta+\log 2\}]}$
		$\displaystyle={\color[rgb]{0.76,0.01,0}\exp[-\exp\{-(x-\alpha-\beta\log 2)/% \beta\}],}$

which is the cdf of a $\operatorname{\mathsf{Gumbel}}(\alpha+\beta\log 2,\beta)$ distribution.