Generalized Distribution of Extreme Values

In the probability theory and in mathematical statistics, extreme value (EV) distribution is a set of probability distributions, developed for theory of extreme values. Generalized distribution of extreme values is a generalization of Gumbel, Frechet and Weibull distributions and is used for approximation modeling of maximum of finite sequences of random variables.

Assume that the random variable X is distributed by the rule described by the function:

Where:

ξ > 0. Form parameter.
. Layout parameter.
. Scale parameter.

Then the random value X has generalized distribution of extreme values or .

Distribution density function:

Distribution characteristics

Mean:

Where γ - Euler constant.

Variance:

Where - gamma function.

Mode:

Median:

Kurtosis coefficient:

Distribution Parameters Estimation

Maximum likelihood estimation

Log-likelihood function for GEV distribution where ξ ≠ 0:

Log-likelihood function for the GEV distribution where ξ = 0:

To find optimal estimation of parameters it is necessary to maximize both likelihood functions and select the maximum result. This is the best estimation of parameters.

Estimation by the method of linear combination of ratios of spaces.