Generalized Pareto distribution
||This article needs additional citations for verification. (March 2012)|
|Parameters|| location (real)
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 . Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as .
The standard GPD is defined by
where the support is for and for . The related location-scale family of distributions is obtained by replacing the argument z by and adjusting the support accordingly.
for when , and when , where , , and .
The probability density function is
again, for , and when .
Special cases 
- 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.
Generating generalized Pareto random variables 
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.
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)
See also 
- Coles, Stuart. An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75.
- 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. "Modelling Excesses over High Thresholds, with an Application". In de Oliveira, J. Tiago. Statistical Extremes and Applications. Kluwer. p. 462.
- Embrechts, Paul; Klüppelberg, Claudia; Mikosch, Thomas. Modelling extremal events for insurance and finance. p. 162.
- Pickands, James (1975), "Statistical inference using extreme order statistics", Annals of Statistics 3: 119–131
- Balkema, A.; Laurens de Haan (1974), "Residual life time at great age", Annals of Probability 2: 792–804
- N. L. Johnson, S. Kotz, and N. Balakrishnan (1994). Continuous Univariate Distributions Volume 1, second edition. New York: Wiley. ISBN 0-471-58495-9. Text "." ignored (help) 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. Text "." ignored (help)
- Arnold, B. C. and Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics. Text "." ignored (help)