Generalized Pareto distribution
Probability density function
 
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]}
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 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
The characteristic and moment generating functions are derived and skewness and kurtosis are obtained from MGF by Muraleedharan and Guedes Soares^{[6]}
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 with scale and shape .
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.
See also
References
 ↑ 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.
Further reading
 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; 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.; Laguna, L. (1977). On generalized Pareto distributions with applications to income data. Ames, Iowa: Iowa State University, Department of Economics.