Generalized extreme value distribution

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. Note that a limit distribution need not exist: this 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.Using the standardized variable s = ( x − μ ) / σ {displaystyle s=(x-mu )/sigma }  , where μ ∈ R {displaystyle mu in mathbb {R} }   is the location parameter and σ > 0 {displaystyle sigma >0}   is the scale parameter, the cumulative distribution function of the GEV distribution isSome simple statistics of the distribution are:The shape parameter ξ {displaystyle xi }   governs the tail behavior of the distribution. The sub-families defined by ξ = 0 {displaystyle xi =0}  , ξ > 0 {displaystyle xi >0}   and ξ < 0 {displaystyle xi <0}   correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are displayed below.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. The generalized extreme value distribution is a special case of a max-stable distribution, and is a transformation of a min-stable distribution.