# Generalized Pareto distribution

In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. It is often used to model the tails of another distribution. It is specified by three parameters: location $\mu$ , scale $\sigma$ , and shape $\xi$ . Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as $\kappa =-\xi \,$ .

Parameters Probability density functionGPD distribution functions for $\mu =0$ and different values of $\sigma$ and $\xi$ Cumulative distribution function $\mu \in (-\infty ,\infty )\,$ location (real) $\sigma \in (0,\infty )\,$ scale (real) $\xi \in (-\infty ,\infty )\,$ shape (real) $x\geqslant \mu \,\;(\xi \geqslant 0)$ $\mu \leqslant x\leqslant \mu -\sigma /\xi \,\;(\xi <0)$ ${\frac {1}{\sigma }}(1+\xi z)^{-(1/\xi +1)}$ where $z={\frac {x-\mu }{\sigma }}$ $1-(1+\xi z)^{-1/\xi }\,$ $\mu +{\frac {\sigma }{1-\xi }}\,\;(\xi <1)$ $\mu +{\frac {\sigma (2^{\xi }-1)}{\xi }}$ ${\frac {\sigma ^{2}}{(1-\xi )^{2}(1-2\xi )}}\,\;(\xi <1/2)$ ${\frac {2(1+\xi ){\sqrt {1-2\xi }}}{(1-3\xi )}}\,\;(\xi <1/3)$ ${\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)$ $\log(\sigma )+\xi +1$ $e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$ $e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$ ## Definition

The standard cumulative distribution function (cdf) of the GPD is defined by

$F_{\xi }(z)={\begin{cases}1-\left(1+\xi z\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-e^{-z}&{\text{for }}\xi =0.\end{cases}}$ where the support is $z\geq 0$ for $\xi \geq 0$ and $0\leq z\leq -1/\xi$ for $\xi <0$ . The corresponding probability density function (pdf) is

$f_{\xi }(z)={\begin{cases}{\frac {1}{\sigma }}(\xi z+1)^{-{\frac {\xi +1}{\xi }}}&{\text{for }}\xi \neq 0,\\{\frac {1}{\sigma }}e^{-z}&{\text{for }}\xi =0.\end{cases}}$ ## Characterization

The related location-scale family of distributions is obtained by replacing the argument z by ${\frac {x-\mu }{\sigma }}$ and adjusting the support accordingly: The cumulative distribution function is

$F_{(\xi ,\mu ,\sigma )}(x)={\begin{cases}1-\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0.\end{cases}}$ for $x\geqslant \mu$ when $\xi \geqslant 0\,$ , and $\mu \leqslant x\leqslant \mu -\sigma /\xi$ when $\xi <0$ , where $\mu \in \mathbb {R}$ , $\sigma >0$ , and $\xi \in \mathbb {R}$ .

The probability density function (pdf) is

$f_{(\xi ,\mu ,\sigma )}(x)={\frac {1}{\sigma }}\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{\left(-{\frac {1}{\xi }}-1\right)}$ ,

again, for $x\geqslant \mu$ when $\xi \geqslant 0$ , and $\mu \leqslant x\leqslant \mu -\sigma /\xi$ when $\xi <0$ .

The pdf is a solution of the following differential equation:

$\left\{{\begin{array}{l}f'(x)(-\mu \xi +\sigma +\xi x)+(\xi +1)f(x)=0,\\f(0)={\frac {\left(1-{\frac {\mu \xi }{\sigma }}\right)^{-{\frac {1}{\xi }}-1}}{\sigma }}\end{array}}\right\}$ ## Special cases

• If the shape $\xi$ and location $\mu$ are both zero, the GPD is equivalent to the exponential distribution.
• With shape $\xi >0$ and location $\mu =\sigma /\xi$ , the GPD is equivalent to the Pareto distribution with scale $x_{m}=\sigma /\xi$ and shape $\alpha =1/\xi$ .
• If $X$ $\sim$ $GPD$ $($ $\mu =0$ , $\sigma$ , $\xi$ $)$ , then $Y=\log(X)$ $\sim$ $exGPD$ $($ $\mu =0$ , $\sigma$ , $\xi$ $)$ , where exGPD stands for the exponentiated generalized Pareto distribution. Unlike GPD, exGPD has the finite moments of all orders and possesses separate interpretations for the scale parameter and the shape parameter, which leads to stable and efficient parameter estimation than using GPD.
• GPD is similar to the Burr distribution.

## Generating generalized Pareto random variables

If U is uniformly distributed on (0, 1], then

$X=\mu +{\frac {\sigma (U^{-\xi }-1)}{\xi }}\sim {\mbox{GPD}}(\mu ,\sigma ,\xi \neq 0)$ and

$X=\mu -\sigma \ln(U)\sim {\mbox{GPD}}(\mu ,\sigma ,\xi =0).$ 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.

### GPD as an Exponential-Gamma Mixture

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

$X|\Lambda \sim Exp(\Lambda )$ and

$\Lambda \sim Gamma(\alpha ,\beta )$ then

$X\sim GPD(\xi =1/\alpha ,\ \sigma =\beta /\alpha )$ Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:$\xi$ must be positive.