Generalized Pareto distribution
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Probability density function
PDF for and different values of and
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 . Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. Some references give the shape parameter as .
The standard cumulative distribution function (cdf) of the GPD is defined by
where the support is for and for .
The cdf of the GPD is a solution of the following differential equation:
The related location-scale 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
again, for when , and when .
The pdf is a solution of the following differential equation:
Characteristic and Moment Generating Functions
- 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 .
GPD is quite similar to the Burr distribution.
Generating generalized Pareto random variables
If U is uniformly distributed on (0, 1], then
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.
GPD as an Exponential-Gamma Mixture
A GPD random variable can also be expressed as an exponential random variable, with a Gamma distributed rate parameter.
Notice however, that since the parameters for the Gamma distribution must be greater than zero, we obtain the additional restrictions that: must be positive.
- Burr distribution
- Pareto distribution
- Generalized extreme value distribution
- Pickands–Balkema–de Haan theorem
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