Generalized Pareto distribution

Parameters Probability density functionGPD distribution functions for ${\displaystyle \mu =0}$ and different values of ${\displaystyle \sigma }$ and ${\displaystyle \xi }$ Cumulative distribution function ${\displaystyle \mu \in (-\infty ,\infty )\,}$ 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 ${\displaystyle \xi <0}$.

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

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

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

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:

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

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 ${\displaystyle \alpha =1/\xi }$.
• If ${\displaystyle X}$ ${\displaystyle \sim }$ ${\displaystyle GPD}$ ${\displaystyle (}$${\displaystyle \mu =0}$, ${\displaystyle \sigma }$, ${\displaystyle \xi }$ ${\displaystyle )}$, then ${\displaystyle Y=\log(X)}$ ${\displaystyle \sim }$ ${\displaystyle exGPD}$ ${\displaystyle (}$${\displaystyle \mu =0}$, ${\displaystyle \sigma }$, ${\displaystyle \xi }$ ${\displaystyle )}$, where exGPD is the exponentiated generalized Pareto distribution. Unlike GPD, exGPD has moments of all orders regardless of its parameter conditions and possesses separate interpretations for the scale parameter and shape parameter, which makes parameter estimation more efficient.
• GPD is 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.

References

1. ^ Coles, Stuart (2001-12-12). An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75. ISBN 9781852334598.
2. ^ Dargahi-Noubary, G. R. (1989). "On tail estimation: An improved method". Mathematical Geology. 21 (8): 829–842. doi:10.1007/BF00894450.
3. ^ Hosking, J. R. M.; Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distribution". Technometrics. 29 (3): 339–349. doi:10.2307/1269343. JSTOR 1269343.
4. ^ Davison, A. C. (1984-09-30). "Modelling Excesses over High Thresholds, with an Application". In de Oliveira, J. Tiago (ed.). Statistical Extremes and Applications. Kluwer. p. 462. ISBN 9789027718044.
5. ^ Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance. p. 162. ISBN 9783540609315.