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

â†‘ Coles, Stuart (2001-12-12). An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75. ISBN 9781852334598.
â†‘ Dargahi-Noubary, G. R. (1989). "On tail estimation: An improved method". Mathematical Geology 21 (8): 829â€“842. doi:10.1007/BF00894450.
â†‘ Hosking, J. R. M.; Wallis, J. R. (1987). "Parameter and Quantile Estimation for the Generalized Pareto Distribution". Technometrics 29 (3): 339â€“349. doi:10.2307/1269343.
â†‘ 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.
â†‘ Embrechts, Paul; KlÃ¼ppelberg, Claudia; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance. p. 162. ISBN 9783540609315.
â†‘ 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): 1861â€“1874. doi:10.9734/JSRR/2014/10087.

References

Pickands, James (1975). "Statistical inference using extreme order statistics". Annals of Statistics 3: 119â€“131. doi:10.1214/aos/1176343003
Balkema, A.; De Haan, Laurens (1974). "Residual life time at great age". Annals of Probability 2 (5): 792â€“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 Zipfâ€“Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conwayâ€“Maxwellâ€“Poisson discrete phase-type Delaporte extended negative binomial Gaussâ€“Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yuleâ€“Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Baldingâ€“Nichols Bates beta beta rectangular Irwinâ€“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 Floryâ€“Schulz FrÃ©chet 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 LÃ©vy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwellâ€“Boltzmann Maxwellâ€“JÃ¼ttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breitâ€“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 Tracyâ€“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 LÃ©vy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Misesâ€“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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