Generalized Pareto Distribution

The probability density function (pdf) of the Generalized Pareto distribution GPD(μ, σ, ξ) with location parameter μ, scale parameter σ > 0, and shape parameter ξ is

Generalized Pareto distribution pdf

The cumulative distribution function is

Generalized Pareto distribution cdf

The pdf and cdf are defined for xμ when ξ ≥ 0 and for μ – σ/ξ  > xμ when ξ < 0.

The inverse of the cdf is

Generalized Pareto distribution inverse

Applications

The Generalized Pareto distribution is used to model the distribution of the tail of another distribution; i.e. the value x ≥ some threshold value μ. The choice of the shape parameter, ξ, depends on the type of distribution whose tail is being modeled. For example

  • ξ = 0 for distributions whose tails decrease exponentially (e.g. the normal distribution)
  • ξ > 0 for distributions whose tails decrease as a polynomial (e.g. the t distribution)
  • ξ < 0 for distributions whose tails are finite (e.g. the beta distribution)

Properties

Figure 1 displays the key properties of the Generalized Pareto distribution.

Generalized Pareto distribution properties

Figure 1 – GPD key properties

Property 1: If ξ > 0 then GPD(m,m/α,1/α) = Pareto distribution with parameters m and α.

Proof: We first note that m/α =mξ. We now match the cdf as follows:

Pareto proof part 1

Pareto proof part 2

Property 2: GPD(0,σ,0) = Exponential distribution with rate parameter 1/σ.

Proof: Similar to the proof of Property 1.

Worksheet Functions

Real Statistics Functions: The Real Statistics Resource Pack provides the following functions.

GPD_DIST(x, μ, σ, ξ, cum) = the pdf of the GPD when cum = FALSE and the corresponding cumulative distribution function when cum = TRUE.

GPD_INV(p, μ, σ, ξ) = the inverse of the GPD at p

Distribution Fitting

Click here for information about how to fit data to a GPD using the method of moments.

References

Mathworks (2022) Generalized Pareto distribution
https://it.mathworks.com/help/stats/generalized-pareto-distribution.html

Wikipedia (2022) Generalized Pareto distribution
https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

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