Generalized Pareto distribution

This article is about a particular family of continuous distributions referred to as the generalized Pareto distribution. For the hierarchy of generalized Pareto distributions, see Pareto distribution.

Generalized Pareto distribution
Probability density function GPD distribution functions for $\mu =0$ and different values of $\sigma$ and $\xi$
Cumulative distribution function
Parameters	$\mu \in (-\infty ,\infty )\,$ location (real) $\sigma \in (0,\infty )\,$ scale (real) $\xi \in (-\infty ,\infty )\,$ shape (real)
Support	$x\geq \mu \,\;(\xi \geq 0)$ $\mu \leq x\leq \mu -\sigma /\xi \,\;(\xi <0)$
PDF	${\frac {1}{\sigma }}(1+\xi z)^{-(1/\xi +1)}$ where $z={\frac {x-\mu }{\sigma }}$
CDF	$1-(1+\xi z)^{-1/\xi }\,$
Mean	$\mu +{\frac {\sigma }{1-\xi }}\,\;(\xi <1)$
Median	$\mu +{\frac {\sigma (2^{\xi }-1)}{\xi }}$
Mode	$\mu$
Variance	${\frac {\sigma ^{2}}{\left(1-\xi \right)^{2}(1-2\xi )}}\,\;(\xi <1/2)$
Skewness	${\frac {2(1+\xi ){\sqrt {1-2\xi }}}{(1-3\xi )}}\,\;(\xi <1/3)$
Excess kurtosis	${\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)$
Entropy	$\log(\sigma )+\xi +1$
MGF	$e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$
CF	$e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$
Method of moments	$\xi ={\frac {1}{2}}\left(1-{\frac {\left(\operatorname {E} [X]-\mu \right)^{2}}{\operatorname {Var} [X]}}\right)$ $\sigma =(\operatorname {E} [X]-\mu )(1-\xi )$

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 $\kappa =-\xi \,$. This parameterization was introduced by James Pickands III .