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 μ {\displaystyle \mu } , scale σ {\displaystyle \sigma } , and shape ξ {\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 κ = − ξ {\displaystyle \kappa =-\xi \,} . This parameterization was introduced by James Pickands III .
Definition The cumulative distribution function of X ∼ GPD ( μ , σ , ξ ) {\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )} ( μ ∈ R {\displaystyle \mu \in \mathbb {R} } , σ > 0 {\displaystyle \sigma >0} , and ξ ∈ R {\displaystyle \xi \in \mathbb {R} } ) is
F μ , σ , ξ ( x ) = { 1 − ( 1 + ξ x − μ σ ) − 1 / ξ for ξ ≠ 0 , 1 − exp ( − x − μ σ ) for ξ = 0 , {\displaystyle F_{\mu ,\sigma ,\xi }(x)={\begin{cases}1-\left(1+\xi {\frac {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}}}
where the support of X is x ≥ μ {\displaystyle x\geq \mu } when ξ ≥ 0 {\displaystyle \xi \geq 0} , and μ ≤ x ≤ μ − σ / ξ {\displaystyle \mu \leq x\leq \mu -\sigma /\xi } when ξ < 0 {\displaystyle \xi <0} . The probability density function (pdf) of X ∼ GPD ( μ , σ , ξ ) {\displaystyle X\sim {\text{GPD}}(\mu ,\sigma ,\xi )} is
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