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 ${\displaystyle \xi }$. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as ${\displaystyle \kappa =-\xi \,}$. This parameterization was introduced by James Pickands III .

The cumulative distribution function of ${\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )}$ (${\displaystyle \mu \in \mathbb {R} }$, ${\displaystyle \sigma >0}$, and ${\displaystyle \xi \in \mathbb {R} }$) is

where the support of X is ${\displaystyle x\geq \mu }$ when ${\displaystyle \xi \geq 0}$, and ${\displaystyle \mu \leq x\leq \mu -\sigma /\xi }$ when ${\displaystyle \xi <0}$.

The probability density function (pdf) of ${\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )}$ is

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

The pdf is a solution of the following differential equation:^{[citation needed]}

The standard cumulative distribution function (cdf) of the GPD is defined using ${\displaystyle z={\frac {x-\mu }{\sigma }}}$ (or, equivalently, setting ${\displaystyle \mu =0}$ and ${\displaystyle \sigma =1}$):

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