GPD: Generalized Pareto distribution

In extraDistr: Additional Univariate and Multivariate Distributions

Description

Density, distribution function, quantile function and random generation for the generalized Pareto distribution.

Usage

dgpd(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pgpd(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgpd(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

rgpd(n, mu = 0, sigma = 1, xi = 0)

Arguments

x, q	vector of quantiles.
mu, sigma, xi	location, scale, and shape parameters. Scale must be positive.
log, log.p	logical; if TRUE, probabilities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x].
p	vector of probabilities.
n	number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x) = [if ξ != 0:] (1+ξ*(x-μ)/σ)^{-(ξ+1)/ξ}/σ [else:] exp(-(x-μ)/σ)/σ

Cumulative distribution function

F(x) = [if ξ != 0:] 1-(1+ξ*(x-μ)/σ)^{-1/ξ} [else:] 1-exp(-(x-μ)/σ)

Quantile function

F^-1(x) = [if ξ != 0:] μ + σ * ((1-p)^{-ξ}-1)/ξ [else:] μ - σ * log(1-p)

References

Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Examples

x <- rgpd(1e5, 5, 2, .1)
hist(x, 100, freq = FALSE, xlim = c(0, 50))
curve(dgpd(x, 5, 2, .1), 0, 50, col = "red", add = TRUE, n = 5000)
hist(pgpd(x, 5, 2, .1))
plot(ecdf(x))
curve(pgpd(x, 5, 2, .1), 0, 50, col = "red", lwd = 2, add = TRUE)

extraDistr documentation built on Sept. 7, 2020, 5:09 p.m.