Generalized Pareto distribution

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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 $\mu \in \R$ is the location parameter, $\sigma >0\,$ the scale parameter and $\xi \in \R$ 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

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)

See also

• Pickands–Balkema–de Haan theorem

References

• 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