# Generalized Pareto distribution

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

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 ${\displaystyle \mu }$, scale ${\displaystyle \sigma }$, and shape ${\displaystyle \xi }$.[1][2] Sometimes it is specified by only scale and shape[3] and sometimes only by its shape parameter. Some references give the shape parameter as ${\displaystyle \kappa =-\xi \,}$.[4]

## Definition

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

${\displaystyle F_{\xi }(z)={\begin{cases}1-\left(1+\xi z\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-e^{-z}&{\text{for }}\xi =0.\end{cases}}}$

where the support is ${\displaystyle z\geq 0}$ for ${\displaystyle \xi \geq 0}$ and ${\displaystyle 0\leq z\leq -1/\xi }$ for ${\displaystyle \xi <0}$.

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

### Differential equation

The cdf of the GPD is a solution of the following differential equation:

${\displaystyle \left\{{\begin{array}{l}(\xi z+1)f_{\xi }'(z)+(\xi +1)f_{\xi }(z)=0,\\f_{\xi }(0)=1\end{array}}\right\}}$

## Characterization

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

${\displaystyle F_{(\xi ,\mu ,\sigma )}(x)={\begin{cases}1-\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }&{\text{for }}\xi \neq 0,\\1-\exp \left(-{\frac {x-\mu }{\sigma }}\right)&{\text{for }}\xi =0.\end{cases}}}$

for ${\displaystyle x\geqslant \mu }$ when ${\displaystyle \xi \geqslant 0\,}$, and ${\displaystyle \mu \leqslant x\leqslant \mu -\sigma /\xi }$ when ${\displaystyle \xi <0}$, where ${\displaystyle \mu \in \mathbb {R} }$, ${\displaystyle \sigma >0}$, and ${\displaystyle \xi \in \mathbb {R} }$.

The probability density function (pdf) is

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

or equivalently

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

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

The pdf is a solution of the following differential equation:

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

## Characteristic and Moment Generating Functions

The characteristic and moment generating functions are derived and skewness and kurtosis are obtained from MGF by Muraleedharan and Guedes Soares[6]

## Special cases

• If the shape ${\displaystyle \xi }$ and location ${\displaystyle \mu }$ are both zero, the GPD is equivalent to the exponential distribution.
• With shape ${\displaystyle \xi >0}$ and location ${\displaystyle \mu =\sigma /\xi }$, the GPD is equivalent to the Pareto distribution with scale ${\displaystyle x_{m}=\sigma /\xi }$ and shape ${\displaystyle \alpha =1/\xi }$.

GPD is quite similar to the Burr distribution.

## Generating generalized Pareto random variables

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

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

and

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

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.

${\displaystyle X|\Lambda \sim Exp(\Lambda )}$

and

${\displaystyle \Lambda \sim Gamma(\alpha ,\beta )}$

then

${\displaystyle X\sim GPD(\xi =1/\alpha ,\ \sigma =\beta /\alpha )}$

Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that:${\displaystyle \xi }$ must be positive.