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The generalised extreme value distribution

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The generalised extreme value (or generalized extreme value) distribution characterises the behaviour of ‘block maxima’ under certain (somewhat restrictive) regularity conditions. See also Nematrian’s webpages about Extreme Value Theory (EVT).

 

[SmartChart][SmartChart][SmartChart]

 

Distribution name

Generalised extreme value (GEV) distribution (for maxima)

Common notation

Parameters

 = shape parameter

 = location parameter

 = scale parameter

Domain

Probability density function

where

Cumulative distribution function

Mean

where  is Euler’s constant, i.e.

Variance

Where

Skewness

where  is the Riemann zeta function, i.e. .

(Excess) kurtosis

Other comments

 defines the tail behaviour of the distribution. The sub-families defined by  (Type I),  (Type II) and  (Type III) correspond to the Gumbel, Frechét and Weibull families respectively.

 

An important special case when analysing threshold exceedances involves  (and normally ) and this special case may be referred to as .

 

Nematrian web functions

 

Functions relating to the above distribution may be accessed via the Nematrian web function library by using a DistributionName of “gev”. For details of other supported probability distributions see here.

 


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