GPD {ReIns}R Documentation

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)

Arguments

x

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

gamma

The γ parameter of the GPD, a real number.

mu

The μ parameter of the GPD, a strictly positive number. Default is 0.

sigma

The σ 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≤ 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.

Details

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

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.

Author(s)

Tom Reynkens.

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")

[Package ReIns version 1.0.8 Index]