Generalized Pareto distribution

σ ∈ ( 0 , ∞ ) {displaystyle sigma in (0,infty ),} scale (real)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 ,} . 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 ,} . The standard cumulative distribution function (cdf) of the GPD is defined by where the support is z ≥ 0 {displaystyle zgeq 0} for ξ ≥ 0 {displaystyle xi geq 0} and 0 ≤ z ≤ − 1 / ξ {displaystyle 0leq zleq -1/xi } for ξ < 0 {displaystyle xi <0} . The related location-scale family of distributions is obtained by replacing the argument z by x − μ σ {displaystyle {frac {x-mu }{sigma }}} and adjusting the support accordingly: The cumulative distribution function is for x ⩾ μ {displaystyle xgeqslant mu } when ξ ⩾ 0 {displaystyle xi geqslant 0,} , and μ ⩽ x ⩽ μ − σ / ξ {displaystyle mu leqslant xleqslant mu -sigma /xi } when ξ < 0 {displaystyle xi <0} , where μ ∈ R {displaystyle mu in mathbb {R} } , σ > 0 {displaystyle sigma >0} , and ξ ∈ R {displaystyle xi in mathbb {R} } . The probability density function (pdf) is again, for x ⩾ μ {displaystyle xgeqslant mu } when ξ ⩾ 0 {displaystyle xi geqslant 0} , and μ ⩽ x ⩽ μ − σ / ξ {displaystyle mu leqslant xleqslant mu -sigma /xi } when ξ < 0 {displaystyle xi <0} . The pdf is a solution of the following differential equation: If U is uniformly distributed on(0, 1], then