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	$e^{\theta\mu}\,\sum_{j=0}^\infty \left[\frac{(\theta\sigma)^j}{\prod_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)$
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 $\xi$ .^[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 $\kappa = - \xi \,$ .^[4]

Definition

The standard cumulative distribution function (cdf) of the GPD is defined by^[5]

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^[6]

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)

Notes

↑ Coles, Stuart (2001-12-12). An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75. ISBN 9781852334598.
↑ Dargahi-Noubary, G. R. (1989). "On tail estimation: An improved method". Mathematical Geology 21 (8): 829–842. doi:10.1007/BF00894450.
↑ 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.
↑ Davison, A. C. (1984-09-30). "Modelling Excesses over High Thresholds, with an Application". In de Oliveira, J. Tiago. Statistical Extremes and Applications. Kluwer. p. 462. ISBN 9789027718044.
↑ Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance. p. 162. ISBN 9783540609315.
↑ Muraleedharan, G.; C, Guedes Soares (2014). "Characteristic and Moment Generating Functions of Generalised Pareto(GP3) and Weibull Distributions". Journal of Scientific Research and Reports 3 (14): 1861–1874. doi:10.9734/JSRR/2014/10087.

References

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, and N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, second edition. New York: Wiley. ISBN 0-471-58495-9. Chapter 20, Section 12: Generalized Pareto Distributions.
Barry C. Arnold (2011). "Chapter 7: Pareto and Generalized Pareto Distributions". In Duangkamon Chotikapanich. Modeling Distributions and Lorenz Curves. New York: Springer. ISBN 9780387727967.
Arnold, B. C. and Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics.

External links

Mathworks: Generalized Pareto distribution

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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