In probability theory and statistics, the generalized extreme value (GEV) distribution[2] is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions.

By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables.

Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables.

Tippett who recognised three different forms outlined below.

However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution.

The origin of the common functional form for all three distributions dates back to at least Jenkinson (1955),[4] though allegedly[3] it could also have been given by von Mises (1936).

, the location parameter, can be any real number, and

, the shape parameter, can be any real number.

The probability density function of the standardized distribution is again valid for

, the density is positive on the whole real line.

is the gamma function, some simple statistics of the distribution are given by:[citation needed] The skewness is The excess kurtosis is:

governs the tail behavior of the distribution.

these correspond, respectively, to the Gumbel, Fréchet, and Weibull families, whose cumulative distribution functions are displayed below.

The subsections below remark on properties of these distributions.

The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima.

A generalised extreme value distribution for data minima can be obtained, for example by substituting

which gives a strictly positive support, in contrast to the use in the formulation of extreme value theory here.

This arises because the ordinary Weibull distribution is used for cases that deal with data minima rather than data maxima.

The distribution here has an addition parameter compared to the usual form of the Weibull distribution and, in addition, is reversed so that the distribution has an upper bound rather than a lower bound.

Importantly, in applications of the GEV, the upper bound is unknown and so must be estimated, whereas when applying the ordinary Weibull distribution in reliability applications the lower bound is usually known to be zero.

Note the differences in the ranges of interest for the three extreme value distributions: Gumbel is unlimited, Fréchet has a lower limit, while the reversed Weibull has an upper limit.

More precisely, univariate extreme value theory describes which of the three is the limiting law according to the initial law  X  and in particular depending on the original distribution's tail.

One can link the type I to types II and III in the following way: If the cumulative distribution function of some random variable

is of type II, and with the positive numbers as support, i.e.

is of type III, and with the negative numbers as support, i.e.

Multinomial logit models, and certain other types of logistic regression, can be phrased as latent variable models with error variables distributed as Gumbel distributions (type I generalized extreme value distributions).

This phrasing is common in the theory of discrete choice models, which include logit models, probit models, and various extensions of them, and derives from the fact that the difference of two type-I GEV-distributed variables follows a logistic distribution, of which the logit function is the quantile function.

The type-I GEV distribution thus plays the same role in these logit models as the normal distribution does in the corresponding probit models.

The cumulative distribution function of the generalized extreme value distribution solves the stability postulate equation.

normally distributed random variables with mean 0 and variance 1.