Generalized Pareto Distribution

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Description

These are the density and random generation functions for the generalized Pareto distribution.

Usage

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dgpd(x, mu, sigma, xi, log=FALSE)
rgpd(n, mu, sigma, xi)

Arguments

x

This is a vector of data.

n

This is a positive scalar integer, and is the number of observations to generate randomly.

mu

This is a scalar or vector location parameter mu. When xi is non-negative, mu must not be greater than x. When xi is negative, mu must be less than x + sigma/xi.

sigma

This is a positive-only scalar or vector of scale parameters sigma.

xi

This is a scalar or vector of shape parameters xi.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

  • Application: Continuous Univariate

  • Density: 1/sigma (1 + xi z)^(-1/xi + 1) where z = (theta - mu)/sigma

  • Inventor: Pickands (1975)

  • Notation 1: theta ~ GPD(mu, sigma, xi)

  • Notation 2: p(theta) ~ GPD(theta | mu, sigma, xi)

  • Parameter 1: location mu, where mu <= theta when xi >= 0, and mu >= theta + sigma/xi when xi < 0

  • Parameter 2: scale sigma > 0

  • Parameter 3: shape xi

  • Mean: mu + sigma / (1 - xi) when xi < 1

  • Variance: sigma^2 / [(1 - xi)^2 (1 - 2 xi)] when xi < 0.5

  • Mode:

The generalized Pareto distribution (GPD) is a more flexible extension of the Pareto (dpareto) distribution. It is equivalent to the exponential distribution when both mu = 0 and xi = 0, and it is equivalent to the Pareto distribution when mu = sigma / xi and xi > 0.

The GPD is often used to model the tails of another distribution, and the shape parameter xi relates to tail-behavior. Distributions with tails that decrease exponentially are modeled with shape xi = 0. Distributions with tails that decrease as a polynomial are modeled with a positive shape parameter. Distributions with finite tails are modeled with a negative shape parameter.

Value

dgpd gives the density, and rgpd generates random deviates.

References

Pickands J. (1975). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics, 3, p. 119–131.

See Also

dpareto

Examples

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library(LaplacesDemon)
x <- dgpd(0,0,1,0,log=TRUE)
x <- rgpd(10,0,1,0)

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