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Probability density function
PDF for and different values of and
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 cumulative distribution function (cdf) of the GPD is defined by
where the support is for and for .
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
again, for when , and when .
The pdf is a solution of the following differential equation:
Characteristic and Moment Generating Functions
- 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 .
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)
I suggest that this paragraph also refer to the link at https://en.wikipedia.org/wiki/Pareto_distribution even though it is mentioned at the top of the page.
- 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.
- 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.
- Pickands, James (1975). "Statistical inference using extreme order statistics". Annals of Statistics 3: 119–131. doi:10.1214/aos/1176343003
- Balkema, A.; De Haan, Laurens (1974). "Residual life time at great age". Annals of Probability 2 (5): 792–804. 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.
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