In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions.

Some references give the shape parameter as

[5] The standard cumulative distribution function (cdf) of the GPD is defined by[6] where the support is

The corresponding probability density function (pdf) is The related location-scale family of distributions is obtained by replacing the argument z by

The probability density function (pdf) of

The pdf is a solution of the following differential equation: [citation needed] If U is uniformly distributed on (0, 1], then and Both formulas are obtained by inversion of the cdf.

In Matlab Statistics Toolbox, you can easily use "gprnd" command to generate generalized Pareto random numbers.

A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter.

and then Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that

In addition to this mixture (or compound) expression, the generalized Pareto distribution can also be expressed as a simple ratio.

and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.

The probability density function(pdf) of

The exGPD has finite moments of all orders for all

participates through the digamma function: Note that for a fixed value for the

plays as the location parameter under the exponentiated generalized Pareto distribution.

Note that the roles of the scale parameter

are separably interpretable, which may lead to a robust efficient estimation for the

(at least up to the second central moment); see the formula of variance

such that its tail distribution is regularly varying with the tail-index

To be specific, the tail distribution is described as It is of a particular interest in the extreme value theory to estimate the shape parameter

is positive (so called the heavy-tailed distribution).

be their conditional excess distribution function.

Pickands–Balkema–de Haan theorem (Pickands, 1975; Balkema and de Haan, 1974) states that for a large class of underlying distribution functions

is well approximated by the generalized Pareto distribution (GPD), which motivated Peak Over Threshold (POT) methods to estimate

: the GPD plays the key role in POT approach.

Technical formulation of the Hill's estimator is as follows.

Then, with this notation, the Hill's estimator (see page 190 of Reference 5 by Embrechts et al [3]) based on the

upper order statistics is defined as In practice, the Hill estimator is used as follows.

Then, select from the set of Hill estimators

: these stable values are regarded as reasonable estimates for the shape parameter