ReIns (version 1.0.10)

GPD: The generalised Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Generalised Pareto Distribution (GPD).

Usage

dgpd(x, gamma, mu = 0, sigma, log = FALSE)
pgpd(x, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE)
qgpd(p, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE)
rgpd(n, gamma, mu = 0, sigma)

Value

dgpd gives the density function evaluated in \(x\), pgpd the CDF evaluated in \(x\) and qgpd the quantile function evaluated in \(p\). The length of the result is equal to the length of \(x\) or \(p\).

rgpd returns a random sample of length \(n\).

Arguments

x

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

gamma

The \(\gamma\) parameter of the GPD, a real number.

mu

The \(\mu\) parameter of the GPD, a strictly positive number. Default is 0.

sigma

The \(\sigma\) parameter of the GPD, a strictly positive number.

log

Logical indicating if the densities are given as \(\log(f)\), default is FALSE.

lower.tail

Logical indicating if the probabilities are of the form \(P(X\le x)\) (TRUE) or \(P(X>x)\) (FALSE). Default is TRUE.

log.p

Logical indicating if the probabilities are given as \(\log(p)\), default is FALSE.

Author

Tom Reynkens.

Details

The Cumulative Distribution Function (CDF) of the GPD for \(\gamma \neq 0\) is equal to \(F(x) = 1-(1+\gamma (x-\mu)/\sigma)^{-1/\gamma}\) for all \(x \ge \mu\) and \(F(x)=0\) otherwise. When \(\gamma=0\), the CDF is given by \(F(x) = 1-\exp((x-\mu)/\sigma)\) for all \(x \ge \mu\) and \(F(x)=0\) otherwise.

References

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

See Also

tGPD, Pareto, EPD, Distributions

Examples

# Plot of the PDF
x <- seq(0, 10, 0.01)
plot(x, dgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="PDF", type="l")

# Plot of the CDF
x <- seq(0, 10, 0.01)
plot(x, pgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="CDF", type="l")