In probability theory and statistics, the generalized extreme value (GEV) distribution 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. that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables. In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after R.A. Fisher and L.H.C. 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 , though allegedly it could also have been given by.

Specification

Using the standardized variable, where \mu, the location parameter, can be any real number, and \sigma > 0 is the scale parameter; the cumulative distribution function of the GEV distribution is then where \xi, the shape parameter, can be any real number. Thus, for \xi > 0, the expression is valid for, while for \xi < 0 it is valid for. In the first case, is the negative, lower end-point, where F is 0 In the special case of x = \mu, we have s = 0, so regardless of the values of \xi and \sigma. The probability density function of the standardized distribution is again valid for in the case \xi > 0, and for in the case \xi < 0. The density is zero outside of the relevant range. In the case \xi = 0, the density is positive on the whole real line. Since the cumulative distribution function is invertible, the quantile function for the GEV distribution has an explicit expression, namely and therefore the quantile density function is valid for \sigma > 0 and for any real \xi.

Summary statistics

Using for where is the gamma function, some simple statistics of the distribution are given by: The skewness is The excess kurtosis is:

Link to Fréchet, Weibull, and Gumbel families

The shape parameter \ \xi\ governs the tail behavior of the distribution. The sub-families defined by three cases: and 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.

Modification for minima rather than maxima

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 \ -x; for ;x; in the distribution function, and subtracting the cumulative distribution from one: That is, replace \ F(x)\ with. Doing so yields yet another family of distributions.

Alternative convention for the Weibull distribution

The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable 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.

Ranges of the distributions

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.

Distribution of log variables

One can link the type I to types II and III in the following way: If the cumulative distribution function of some random variable \ X\ is of type II, and with the positive numbers as support, i.e. then the cumulative distribution function of \ln X is of type I, namely Similarly, if the cumulative distribution function of \ X\ is of type III, and with the negative numbers as support, i.e. then the cumulative distribution function of \ \ln (-X)\ is of type I, namely

Link to logit models (logistic regression)

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.

Properties

The cumulative distribution function of the generalized extreme value distribution solves the stability postulate equation. The generalized extreme value distribution is a special case of a max-stable distribution, and is a transformation of a min-stable distribution.

Applications

Example for Normally distributed variables

Let be i.i.d. normally distributed random variables with mean 0 and variance 1 . The Fisher–Tippett–Gnedenko theorem tells us that where This allow us to estimate e.g. the mean of from the mean of the GEV distribution: where is the Euler–Mascheroni constant.

Related distributions

Proofs

4. Let then the cumulative distribution of is:
5. Let then the cumulative distribution of is: