Generalized Pareto distribution

Generalized Pareto distribution
	Probability density function GPD distribution functions for and different values of and
	Cumulative distribution function
Parameters	location (real); scale (real); shape (real)
Support	;
PDF	; where
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy
MGF
CF

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}}

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\}

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 $\alpha=1/\xi$ .
If $X$ $\sim$ $GPD$ $($ $\mu =0$ , $\sigma$ , $\xi$ $)$ , then $Y=\log(X)$ $\sim$ $exGPD$ $($ $\mu =0$ , $\sigma$ , $\xi$ $)$ , 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 quite similar to the Burr distribution.

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.

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.

X|\Lambda \sim Exp(\Lambda )

and

\Lambda \sim Gamma(\alpha ,\beta )

then

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: $\xi$ must be positive.

References

↑ 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.

External links

Mathworks: Generalized Pareto distribution

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[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.

[4] 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.

[5] Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas (1997-01-01). Modelling extremal events for insurance and finance. p. 162. ISBN 9783540609315.

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 Lomax 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 Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace 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