name =Generalized extreme value
parameters = location (real)
In probability theory and statistics, the generalized extreme value distribution (GEV) 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. Its importance arises from the fact that it is the limit distribution of the maxima of a sequence of independent and identically distributed random variables. Because of this, the GEV is used as an approximation to model the maxima of long (finite) sequences of random variables.
The generalized extreme value distribution has cumulative distribution function
for , where is the location parameter, the scale parameter and the shape parameter.
The density function is, consequently
again, for .
Mean, standard deviation, mode, skewness and kurtosis excess
The skewness is:
The kurtosis excess is::
where , k=1,2,3,4, and is Gamma function.
Link to Fréchet, Weibull and Gumbel families
The shape parameter governs the tail behaviour of the distribution. The sub-families defined by , and correspond, respectively, to the Gumbel, Fréchet and Weibull families, whose cumulative distribution functions are displayed below.
* Gumbel or type I extreme value distribution:
* Fréchet or type II extreme value distribution:
* Reversed Weibull or type III extreme value distribution: