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	${\displaystyle }$
Variance	${\frac {\sigma ^{2}}{(1-\xi )^{2}(1-2\xi )}}\,\;(\xi <1/2)$
Skewness	${\frac {2(1+\xi ){\sqrt {(}}1-{2\xi })}{(1-3\xi )}}\,\;(\xi <1/3)$
Excess kurtosis	${\frac {3(1-2\xi )(2\xi ^{2}+\xi +3)}{(1-3\xi )(1-4\xi )}}-3\,\;(\xi <1/4)$
Entropy	${\displaystyle }$
MGF	$e^{\theta \mu }\,\sum _{j=0}^{\infty }\left[{\frac {(\theta \sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$
CF	$e^{it\mu }\,\sum _{j=0}^{\infty }\left[{\frac {(it\sigma )^{j}}{\prod _{k=0}^{j}(1-k\xi )}}\right],\;(k\xi <1)$

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 $\mu$, 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.

See also

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

Further reading

• Pickands, James (1975). "Statistical inference using extreme order statistics". Annals of Statistics. 3: 119–131. doi:10.1214/aos/1176343003.
• Balkema, A.; De Haan, Laurens (1974). "Residual life time at great age". Annals of Probability. 2 (5): 792–804. doi:10.1214/aop/1176996548.
• N. L. Johnson; S. Kotz; N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, second edition. New York: Wiley. ISBN 0-471-58495-9. {{cite book}}: Invalid |ref=harv (help) Chapter 20, Section 12: Generalized Pareto Distributions.
• Barry C. Arnold (2011). "Chapter 7: Pareto and Generalized Pareto Distributions". In Duangkamon Chotikapanich (ed.). Modeling Distributions and Lorenz Curves. New York: Springer. ISBN 9780387727967. {{cite book}}: Invalid |ref=harv (help)
• Arnold, B. C.; Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics. {{cite book}}: Invalid |ref=harv (help)

External links

• Mathworks: Generalized Pareto distribution