Generalized Pareto distribution

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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 Generalized Pareto.svg
PDF for and different values of and

location (real)
scale (real)

shape (real)


Ex. kurtosis

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 , scale , and shape .[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 .[4]


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

where the support is for and for .

Differential equation[edit]

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


The related location-scale family of distributions is obtained by replacing the argument z by and adjusting the support accordingly: The cumulative distribution function is

for when , and when , where , , and .

The probability density function (pdf) is


or equivalently


again, for when , and when .

The pdf is a solution of the following differential equation:

Characteristic and Moment Generating Functions[edit]

The characteristic and moment generating functions are derived and skewness and kurtosis are obtained from MGF by Muraleedharan and Guedes Soares[6]

Special cases[edit]

  • If the shape and location are both zero, the GPD is equivalent to the exponential distribution.
  • With shape and location , the GPD is equivalent to the Pareto distribution with scale and shape .

GPD is quite similar to the Burr distribution.

Generating generalized Pareto random variables[edit]

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


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[edit]

A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter.



Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that: must be positive.

See also[edit]


  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. 
  6. ^ 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. 

Further reading[edit]

External links[edit]