Generalized Pareto distribution

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This article is about a particular family of continuous distributions referred to as the generalized Pareto distribution. For the hierarchy of generalized Pareto distributions, see Pareto distribution.
Generalized Pareto distribution
Probability density function
PDF Generalized Pareto.svg
PDF for \mu=0 and different values of \sigma and \xi
Parameters

\mu \in (-\infty,\infty) \, location (real)
\sigma \in (0,\infty)    \, scale (real)

\xi\in (-\infty,\infty)  \, shape (real)
Support

x \geqslant \mu\,\;(\xi \geqslant 0)

\mu \leqslant x \leqslant \mu-\sigma/\xi\,\;(\xi < 0)
PDF

\frac{1}{\sigma}(1 + \xi z )^{-(1/\xi +1)}

where z=\frac{x-\mu}{\sigma}
CDF 1-(1+\xi z)^{-1/\xi} \,
Mean \mu + \frac{\sigma}{1-\xi}\, \; (\xi < 1)
Median \mu + \frac{\sigma( 2^{\xi} -1)}{\xi}
Mode
Variance \frac{\sigma^2}{(1-\xi)^2(1-2\xi)}\, \; (\xi < 1/2)
Skewness \frac{2(1+\xi)\sqrt(1-{2\xi})}{(1-3\xi)}\,\;(\xi<1/3)
Ex. kurtosis \frac{3(1-2\xi)(2\xi^2+\xi+3)}{(1-3\xi)(1-4\xi)}-3\,\;(\xi<1/4)
Entropy
MGF e^{\theta\mu}\,\sum_{j=0}^\infty \left[\frac{(\theta\sigma)^j}{\pi_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)
CF e^{it\mu}\,\sum_{j=0}^\infty \left[\frac{(it\sigma)^j}{\pi_{k=0}^j(1-k\xi)}\right], \;(k\xi<1)

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 \mu, scale \sigma, and shape \xi.[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  \kappa =  - \xi \,.[4]

Definition[edit]

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

F_{\xi}(z) = \begin{cases}
1 - \left(1+ \xi z\right)^{-1/\xi} & \text{for }\xi \neq 0, \\
1 - e^{-z} & \text{for }\xi = 0.
\end{cases}

where the support is  z \geq 0 for  \xi \geq 0 and  0 \leq z \leq - 1 /\xi for  \xi < 0.

f_{\xi}(z) = \begin{cases}
(\xi  z+1)^{-\frac{\xi +1}{\xi }} & \text{for }\xi \neq 0, \\
e^{-z} & \text{for }\xi = 0.
\end{cases}

Differential equation[edit]

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

\left\{\begin{array}{l}
(\xi  z+1) f_{\xi}'(z)+(\xi +1) f_{\xi}(z)=0, \\
f_{\xi}(0)=1
\end{array}\right\}

Characterization[edit]

The related location-scale family of distributions is obtained by replacing the argument z by \frac{x-\mu}{\sigma} and adjusting the support accordingly: The cumulative distribution function is

F_{(\xi,\mu,\sigma)}(x) = \begin{cases}
1 - \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi} & \text{for }\xi \neq 0, \\
1 - \exp \left(-\frac{x-\mu}{\sigma}\right) & \text{for }\xi = 0.
\end{cases}

for  x \geqslant \mu when  \xi \geqslant 0 \,, and  \mu \leqslant x \leqslant \mu - \sigma /\xi when  \xi < 0, where \mu\in\mathbb R, \sigma>0, and \xi\in\mathbb R.

The probability density function (pdf) is

f_{(\xi,\mu,\sigma)}(x) = \frac{1}{\sigma}\left(1 + \frac{\xi (x-\mu)}{\sigma}\right)^{\left(-\frac{1}{\xi} - 1\right)},

or equivalently

f_{(\xi,\mu,\sigma)}(x) = \frac{\sigma^{\frac{1}{\xi}}}{\left(\sigma + \xi (x-\mu)\right)^{\frac{1}{\xi}+1}},

again, for  x \geqslant \mu when  \xi \geqslant 0, and  \mu \leqslant x \leqslant \mu - \sigma /\xi when  \xi < 0.

The pdf is a solution of the following differential equation:

\left\{\begin{array}{l}
f'(x) (-\mu \xi +\sigma+\xi x)+(\xi+1) f(x)=0, \\
f(0)=\frac{\left(1-\frac{\mu \xi}{\sigma}\right)^{-\frac{1}{\xi }-1}}{\sigma}
\end{array}\right\}

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]

Generating generalized Pareto random variables[edit]

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

 X = \mu + \frac{\sigma (U^{-\xi}-1)}{\xi} \sim \mbox{GPD}(\mu, \sigma, \xi \neq 0)

and

 X = \mu - \sigma \ln(U) \sim \mbox{GPD}(\mu,\sigma,\xi =0).

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]

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.

Notes[edit]

  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.  edit
  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.  edit
  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. 

References[edit]

  • 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.
  • 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]