Generalized Extreme Value Distribution

The Generalized Extreme Value (GEV) distribution is a family of continuous probability distributions developed to model the maximum or minimum of a large collection of random observations. It plays a crucial role in extreme value theory and is widely used in fields such as hydrology, finance, insurance, environmental science, and engineering for risk assessment and management.

Definition

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The GEV distribution encompasses three types of extreme value distributions: Gumbel, Fréchet, and Weibull (also known as Type I, II, and III). Its cumulative distribution function (CDF) is given by:

where:

is the location parameter,
is the scale parameter,
is the shape parameter.

The shape parameter determines the type of distribution:

: Fréchet (Type II)
: Gumbel (Type I)
: Weibull (Type III)