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**Generalized Pareto**
Probability density function
Cumulative distribution function
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
Kurtosis
Entropy
mgf
Char. func.

The family of generalized Pareto distributions (GPD) has three parameters $\mu,\sigma \,$ and $\xi \,$ .

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 $\mu\in\mathbb R$ is the location parameter, $\sigma>0 \,$ the scale parameter and $\xi\in\mathbb R$ the shape parameter. Note that some references give the "shape parameter" as $\kappa = - \xi \,$ .

The probability density function is:

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

or

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

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

Generating generalized Pareto random variables Edit

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

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

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)

ReferencesEdit

Balkema, A.; Laurens de Haan (1974), "Residual life time at great age", Annals of Probability 2: 792–804, http://projecteuclid.org/euclid.aop/1176996548
Pickands, James (1975), "Statistical inference using extreme order statistics", Annals of Statistics 3: 119–131, http://projecteuclid.org/euclid.aos/1176343003

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v·d·e Some common univariate probability distributions
Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull
Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson
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Generalized Pareto distribution

Generating generalized Pareto random variables Edit

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