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Probability density function
PDF for and different values of and 

Parameters  shape (real) 

Support 



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 , 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]}
Contents
Definition[edit]
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:
Characterization[edit]
The related locationscale 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 .
Generating generalized Pareto random variables[edit]
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.
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[edit]
Notes[edit]
 ^ Coles, Stuart (20011212). An Introduction to Statistical Modeling of Extreme Values. Springer. p. 75. ISBN 9781852334598.
 ^ DargahiNoubary, 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. (19840930). "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 (19970101). 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.
References[edit]
 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 0471584959. 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.
External links[edit]

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