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 is

Standard Gumbel distribution

The standard Gumbel distribution is the case where \mu = 0 and \beta = 1 with cumulative distribution function and probability density function In this case the mode is 0, the median is, the mean is (the Euler–Mascheroni constant), and the standard deviation is The cumulants, for n > 1, are given by

Properties

The mode is μ, while the median is and the mean is given by where \gamma is the Euler–Mascheroni constant. The standard deviation \sigma is hence At the mode, where x = \mu, the value of becomes, irrespective of the value of \beta. If G_1,...,G_k are iid Gumbel random variables with parameters (\mu,\beta) then is also a Gumbel random variable with parameters. If are iid random variables such that has the same distribution as G_1 for all natural numbers k, then G_1 is necessarily Gumbel distributed with scale parameter \beta (actually it suffices to consider just two distinct values of k>1 which are coprime).

Related distributions

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

Occurrence and applications

Gumbel has shown that the maximum value (or last order statistic) in a sample of random variables following an exponential distribution minus the natural logarithm of the sample size approaches the Gumbel distribution as the sample size increases. Concretely, let be the probability distribution of x and its cumulative distribution. Then the maximum value out of N realizations of x is smaller than X if and only if all realizations are smaller than X. So the cumulative distribution of the maximum value \tilde{x} satisfies and, for large N, the right-hand-side converges to In hydrology, therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes, and also to describe droughts. Gumbel has also shown that the estimator r/(n+1) for the probability of an event — where r is the rank number of the observed value in the data series and n is the total number of observations — is an unbiased estimator of the cumulative probability around the mode of the distribution. Therefore, this estimator is often used as a plotting position. In number theory, the Gumbel distribution approximates the number of terms in a random partition of an integer as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations. It appears in the coupon collector's problem.

Gumbel reparameterization tricks

In machine learning, the Gumbel distribution is sometimes employed to generate samples from the categorical distribution. This technique is called "Gumbel-max trick" and is a special example of "reparameterization tricks". In detail, let be nonnegative, and not all zero, and let be independent samples of Gumbel(0, 1), then by routine integration,That is, Equivalently, given any, we can sample from its Boltzmann distribution by Related equations include:

Random variate generation

Since the quantile function (inverse cumulative distribution function), Q(p), of a Gumbel distribution is given by the variate Q(U) has a Gumbel distribution with parameters \mu and \beta when the random variate U is drawn from the uniform distribution on the interval (0,1).

Probability paper

In pre-software times probability paper was used to picture the Gumbel distribution (see illustration). The paper is based on linearization of the cumulative distribution function F : In the paper the horizontal axis is constructed at a double log scale. The vertical axis is linear. By plotting F on the horizontal axis of the paper and the x-variable on the vertical axis, the distribution is represented by a straight line with a slope 1/\beta. When distribution fitting software like CumFreq became available, the task of plotting the distribution was made easier.