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. 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. 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.
Definitions The cumulative distribution function of the Gumbel distribution (maximum case) is
F ( x ; μ , β ) = e − e − ( x − μ ) / β {\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}\,}
Standard Gumbel distribution The standard Gumbel distribution is the case where μ = 0 {\displaystyle \mu =0} and β = 1 {\displaystyle \beta =1} with cumulative distribution function
F ( x ) = e − e − x {\displaystyle F(x)=e^{-e^{-x}}\,}
and probability density function
f ( x ) = e − ( x + e − x ) . {\displaystyle f(x)=e^{-(x+e^{-x})}.}
In this case the mode is 0, the median is − ln ( ln ( 2 ) ) ≈ 0.3665 {\displaystyle -\ln(\ln(2))\approx 0.3665} , the mean is γ ≈ 0.5772 {\displaystyle \gamma \approx 0.5772} (the Euler–Mascheroni constant), and the standard deviation is π / 6 ≈ 1.2825. {\displaystyle \pi /{\sqrt {6}}\approx 1.2825.}
The cumulants, for n > 1, are given by
κ n = ( n − 1 ) ! ζ ( n ) . {\displaystyle \kappa _{n}=(n-1)!\zeta (n).}
Properties The mode is μ {\textstyle \mu } , while the median is μ − β ln ( ln 2 ) , {\displaystyle \mu -\beta \ln \left(\ln 2\right),} and the mean is given by
E ( X ) = μ + γ β {\displaystyle \operatorname {E} (X)=\mu +\gamma \beta } , where γ {\displaystyle \gamma } is the Euler–Mascheroni constant. The standard deviation σ {\displaystyle \sigma } is β π / 6 {\displaystyle \beta \pi /{\sqrt {6}}} hence β = σ 6 / π ≈ 0.78 σ . {\displaystyle \beta =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .} At the mode, where x = μ {\displaystyle x=\mu } , the value of F ( x ; μ , β ) {\displaystyle F(x;\mu ,\beta )} becomes e − 1 ≈ 0.37 {\displaystyle e^{-1}\approx 0.37} , irrespective of the value of β . {\displaystyle \beta .}
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