Gumbel distribution

In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.

Notation

${\text{Gumbel}}(\mu ,\beta )$

$\mu ,$ location (real)
$\beta >0,$ scale (real)

$x\in \mathbb {R}$

${\frac {1}{\beta }}e^{-(z+e^{-z})}$
where $z={\frac {x-\mu }{\beta }}$

$e^{-e^{-(x-\mu )/\beta }}$

$\mu +\beta \gamma$
where $\gamma$ is the Euler–Mascheroni constant

$\mu -\beta \ln(\ln 2)$

$\mu$

${\frac {\pi ^{2}}{6}}\beta ^{2}$

${\frac {12{\sqrt {6}}\,\zeta (3)}{\pi ^{3}}}\approx 1.14$

${\frac {12}{5}}$

$\ln(\beta )+\gamma +1$

$\Gamma (1-\beta t)e^{\mu t}$

$\Gamma (1-i\beta t)e^{i\mu t}$

This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory, which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type.^[a]

The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution). It is related to the Gompertz distribution: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.

In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed random variables has a logistic distribution.

The Gumbel distribution is named after Emil Julius Gumbel (1891–1966), based on his original papers describing the distribution.^[1]^[2]

If $X$ has a Gumbel distribution, then the conditional distribution of Y = −X given that Y is positive, or equivalently given that X is negative, has a . The cdf G of Y is related to F, the cdf of X, by the formula $G(y)=P(Y\leq y)=P(X\geq -y\mid X\leq 0)=(F(0)-F(-y))/F(0)$ for y > 0. Consequently, the densities are related by $g(y)=f(-y)/F(0)$ : the Gompertz density is proportional to a reflected Gumbel density, restricted to the positive half-line.^[4]

Gompertz distribution

If X is an exponentially distributed variable with mean 1, then −log(X) has a standard Gumbel distribution.

If $X\sim \mathrm {Gumbel} (\alpha _{X},\beta )$ and $Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )$ are independent, then $X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,$ (see ).

Logistic distribution

If $X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )$ are independent, then $X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )$ . Note that $E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)$ . More generally, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.

[5]

Theory related to the generalized multivariate log-gamma distribution provides a multivariate version of the Gumbel distribution.

If $x\sim \operatorname {Exp} (\lambda )$ , then $(-\ln x-\gamma )\sim {\text{Gumbel}}(-\gamma +\ln \lambda ,1)$ .

$\arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}$ .

$\max _{i}(g_{i}+\log \pi _{i})\sim {\text{Gumbel}}\left(\log \left(\sum _{i}\pi _{i}\right),1\right)$ . That is, the Gumbel distribution is a max-stable distribution family.

$\mathbb {E} [\max _{i}(g_{i}+\beta x_{i})]=\log \left(\sum _{i}e^{\beta x_{i}}\right)+\gamma .$

Gumbel distribution

Notation

Notation

Parameters

Support

PDF

CDF

Mean

Median

Mode

Variance

Skewness

Excess kurtosis

Entropy

MGF

CF

Gompertz distribution

Logistic distribution

[5]

Type-2 Gumbel distribution

Extreme value theory

Generalized extreme value distribution

Fisher–Tippett–Gnedenko theorem

Emil Julius Gumbel