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 PDF for $\mu=0$ and different values of $\sigma$ and $\xi$
Parameters	$\mu \in (-\infty,\infty) \,$ location (real) $\sigma \in (0,\infty) \,$ scale (real) $\xi\in (-\infty,\infty) \,$ shape (real)
Support	$x \geqslant \mu\,\;(\xi \geqslant 0)$ $\mu \leqslant x \leqslant \mu-\sigma/\xi\,\;(\xi < 0)$
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
Variance	$\frac{\sigma^2}{(1-\xi)^2(1-2\xi)}\, \; (\xi < 1/2)$
Skewness	$\frac{2(1+\xi)\sqrt(1-{2\xi})}{(1-3\xi)}\,\;(\xi<1/3)$
Ex. kurtosis	$\frac{3(1-2\xi)(2\xi^2+\xi+3)}{(1-3\xi)(1-4\xi)}-3\,\;(\xi<1/4)$
Entropy
MGF	$e^{\theta\mu}\,\sum_{j=0}^\infty \left[\frac{(\theta\sigma)^j}{\pi_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)$
CF	$e^{it\mu}\,\sum_{j=0}^\infty \left[\frac{(it\sigma)^j}{\pi_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)$

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 $\mu$, 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 $\kappa = - \xi \,$.

Definition

The standard cumulative distribution function (cdf) of the GPD is defined by

$F_{\xi}(z) = \begin{cases} 1 - \left(1+ \xi z\right)^{-1/\xi} & \text{for }\xi \neq 0, \\ 1 - e^{-z} & \text{for }\xi = 0. \end{cases}$

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

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

Differential equation

The cdf of the GPD is a solution of the following differential equation:

$\left\{\begin{array}{l} (\xi z+1) f_{\xi}'(z)+(\xi +1) f_{\xi}(z)=0, \\ f_{\xi}(0)=1 \end{array}\right\}$

Characterization

The related location-scale family of distributions is obtained by replacing the argument z by $\frac{x-\mu}{\sigma}$ and adjusting the support accordingly: The cumulative distribution function is

$F_{(\xi,\mu,\sigma)}(x) = \begin{cases} 1 - \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi} & \text{for }\xi \neq 0, \\ 1 - \exp \left(-\frac{x-\mu}{\sigma}\right) & \text{for }\xi = 0. \end{cases}$

for $x \geqslant \mu$ when $\xi \geqslant 0 \,$, and $\mu \leqslant x \leqslant \mu - \sigma /\xi$ when $\xi < 0$, where $\mu\in\mathbb R$, $\sigma>0$, and $\xi\in\mathbb R$.

The probability density function (pdf) is

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

or equivalently

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

again, for $x \geqslant \mu$ when $\xi \geqslant 0$, and $\mu \leqslant x \leqslant \mu - \sigma /\xi$ when $\xi < 0$.

The pdf is a solution of the following differential equation:

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

Characteristic and Moment Generating Functions

The characteristic and moment generating functions are derived and skewness and kurtosis are obtained from MGF by Muraleedharan and Guedes Soares

Special cases

• If the shape $\xi$ and location $\mu$ are both zero, the GPD is equivalent to the exponential distribution.
• With shape $\xi > 0$ and location $\mu = \sigma/\xi$, the GPD is equivalent to the Pareto distribution with scale $x_m=\sigma/\xi$ and shape $\alpha=1/\xi$.

Generating generalized Pareto random variables

If U is uniformly distributed on (0, 1], then

$X = \mu + \frac{\sigma (U^{-\xi}-1)}{\xi} \sim \mbox{GPD}(\mu, \sigma, \xi \neq 0)$

and

$X = \mu - \sigma \ln(U) \sim \mbox{GPD}(\mu,\sigma,\xi =0).$

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.

With GNU R you can use the packages POT or evd with the "rgpd" command (see for exact usage: http://rss.acs.unt.edu/Rdoc/library/POT/html/simGPD.html)

See also

• Pareto_distribution
• Generalized extreme value distribution
• Pickands–Balkema–de Haan theorem

References

• Chapter 20, Section 12: Generalized Pareto Distributions.

External links

• Mathworks: Generalized Pareto distribution