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The generalised Pareto distribution

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The generalised Pareto distribution (generalized Pareto distribution) arises in Extreme Value Theory (EVT). If the relevant regularity conditions are satisfied then the tail of a distribution (above some suitably high threshold), i.e. the distribution of ‘threshold exceedances’, tends to a generalized Pareto distribution.

 

Care is needed with EVT because what we are in effect doing with it is to extrapolate into the tail of the distribution. Extrapolation is an intrinsically imprecise and subjective mathematical activity. We can in effect view the regularity conditions that need to be satisfied if EVT applies as corresponding to requiring that this extrapolation is done in a particular manner.

 

[SmartChart][SmartChart][SmartChart]

 

Distribution name

Generalised Pareto distribution (GPD)

Common notation

Parameters

 = shape parameter

 = location parameter

 = scale parameter ()

Domain

Probability density function

where

Cumulative distribution function

Mean

Variance

Skewness

(Excess) kurtosis

Other comments

If  is uniformly distributed,  then the variable .

 

The mean excess function for a GPD, i.e.  takes a particularly simple form which is linear in , i.e.

 

Nematrian web functions

 

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

 


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