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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\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)

See alsoEdit

ReferencesEdit

Bvn-small Probability distributions [[[:Template:Tnavbar-plain-nodiv]]]
Univariate Multivariate
Discrete: BernoullibinomialBoltzmanncompound PoissondegeneratedegreeGauss-Kuzmingeometrichypergeometriclogarithmicnegative binomialparabolic fractalPoissonRademacherSkellamuniformYule-SimonzetaZipfZipf-Mandelbrot Ewensmultinomial
Continuous: BetaBeta primeCauchychi-squareDirac delta functionErlangexponentialexponential powerFfadingFisher's zFisher-TippettGammageneralized extreme valuegeneralized hyperbolicgeneralized inverse GaussianHotelling's T-squarehyperbolic secanthyper-exponentialhypoexponentialinverse chi-squareinverse gaussianinverse gammaKumaraswamyLandauLaplaceLévyLévy skew alpha-stablelogisticlog-normalMaxwell-BoltzmannMaxwell speednormal (Gaussian)ParetoPearsonpolarraised cosineRayleighrelativistic Breit-WignerRiceStudent's ttriangulartype-1 Gumbeltype-2 GumbeluniformVoigtvon MisesWeibullWigner semicircle DirichletKentmatrix normalmultivariate normalvon Mises-FisherWigner quasiWishart
Miscellaneous: Cantorconditionalexponential familyinfinitely divisiblelocation-scale familymarginalmaximum entropy phase-typeposterior priorquasisampling
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