Generalized extreme value distribution | Vose Software

# Generalized extreme value distribution

Format: GEV(a,b,c)

The generalized extreme value (GEV) distribution is a continuous probability distribution developed within extreme value theory. It combines the Gumbel, Fréchet and Weibull extreme value distributions. From extreme value theory, the GEV distribution is the limit distribution of normalized maxima of a sequence of independent and identically distributed random variables.

The parameter a controls the location;

The parameter b controls the scale;

The parameter c controls the shape.

The GEV distribution is equivalent to a Gumbel, Fréchet or Weibull distribution dependent on whether c = 0, c > 0 or c < 0 respectively.

## Uses

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

The generalized extreme value distribution is sometimes known as the Fisher–Tippett distribution.

The ordinary Weibull distribution arises in reliability applications and is obtained from the distribution here by using the variable y = c − x, which gives a strictly positive support - in contrast to the use in the extreme value theory here. This arises because the Weibull distribution is used in cases that deal with the minimum rather than the maximum. 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 while when applying the Weibull distribution the lower bound is known to be zero.

## ModelRisk functions added to Microsoft Excel for the Generalized Extreme Value (GEV) distribution

VoseGEV generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseGEVObject constructs a distribution object for this distribution.

VoseGEVProb returns the probability density or cumulative distribution function for this distribution.

VoseGEVProb10 returns the log10 of the probability density or cumulative distribution function.

VoseGEVFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseGEVFitObject constructs a distribution object of this distribution fitted to data.

VoseGEVFitP returns the parameters of this distribution fitted to data.