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Wikipedia / Generalized Pareto distribution

In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions. Sometimes it is specified by only scale and shape. This parameterization was introduced by James Pickands III .

Definition

The cumulative distribution function) is

where the support of .

The probability density function is

.

, is the solution to the following nonlinear ordinary differential equation:

  1. .

is associated with the linear source of uncertainty.

):

.

The corresponding probability density function (pdf) is

Related distributions

(flipped power function distribution).

Prediction

  • It is often of interest to predict probabilities of out-of-sample data under the assumption that both the training data and the out-of-sample data follow a GPD.

  • Predictions of probabilities generated by substituting maximum likelihood estimates of the GPD parameters into the cumulative distribution function ignore parameter uncertainty. As a result, the probabilities are not well calibrated, do not reflect the frequencies of out-of-sample events, and, in particular, underestimate the probabilities of out-of-sample tail events.

  • Predictions generated using the objective Bayesian approach of calibrating prior prediction have been shown to greatly reduce this underestimation, although not completely eliminate it. Calibrating prior prediction is implemented in the R software package fitdistcp.

Generating generalized Pareto random variables

Generating GPD random variables

If U is uniformly distributed on , then

Both formulas are obtained by inversion of the cdf.

The Pareto package in R and the gprnd command in the Matlab Statistics Toolbox can be used 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

.

must be positive.

and taking into account that the rate parameters of the exponential and gamma distribution are simply inverse multiplicative constants.

Exponentiated generalized Pareto distribution

The exponentiated generalized Pareto distribution (exGPD)

is distributed according to the exponentiated generalized Pareto distribution.

The probability density function is

.

is positive.

The exGPD.

The moment-generating function denote the beta function and gamma function, respectively.

The expected value participates through the digamma function plays as the location parameter under the exponentiated generalized Pareto distribution.

The variance only through the polygamma function of order 1 (also called the trigamma function.

wherein both parameters are participated.

The Hill estimator

observations (which need not be i.i.d.) from an unknown heavy-tailed distribution It is of a particular interest in extreme value theory is positive (so-called heavy-tailed distribution).

be their conditional excess distribution function. The Pickands–Balkema–de Haan theorem: the GPD plays the key role in the POT approach.

A renowned estimator using the POT methodology is the Hill estimator. With this notation, the Hill estimator. Note that the Hill estimator. (The Pickands estimator also employs the log-transformation, but in a slightly different way.)

See also

References

Further reading

  • Chapter 20, Section 12: Generalized Pareto Distributions.