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\gamma parameter of the GPD, a real number.

mu

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

sigma

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

log

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

lower.tail

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

log.p

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

Details

The Cumulative Distribution Function (CDF) of the GPD for γ0\gamma \neq 0 is equal to F(x)=1(1+γ(xμ)/σ)1/γF(x) = 1-(1+\gamma (x-\mu)/\sigma)^{-1/\gamma} for all xμx \ge \mu and F(x)=0F(x)=0 otherwise. When γ=0\gamma=0, the CDF is given by F(x)=1exp((xμ)/σ)F(x) = 1-\exp((x-\mu)/\sigma) for all xμx \ge \mu and F(x)=0F(x)=0 otherwise.

Value

dgpd gives the density function evaluated in xx, pgpd the CDF evaluated in xx and qgpd the quantile function evaluated in pp. The length of the result is equal to the length of xx or pp.

rgpd returns a random sample of length nn.

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 1.0.14 Index]