# Generalized Pareto distribution

Parameters Probability density functionPDF for and different values of and location (real) scale (real) shape (real) where

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]

## Definition

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

where the support is for and for .

### Differential equation

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

## Characterization

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

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

## Generating generalized Pareto random variables

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

and

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.

## References

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.