gpd: Generalised Pareto distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Generalised Pareto distribution

Description

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

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	location parameter, the threshold below which the density is zero.
sigma	scale parameter, must be positive.
xi	shape parameter (real).
log, log.p	logical; if TRUE, probabilities/ densities p are returned as \log(p).
lower.tail	logical; if TRUE (default), probabilities are P[X \le x], otherwise P[X > x].
p	vector of probabilities
n	number of random values to return

Details

dgpd and pgpd allow for automatic differentiation with RTMB.

With z = (x - \mu) / \sigma the survival function is

1 - F(x;\,\mu,\sigma,\xi) = \begin{cases} (1 + \xi z)^{-1/\xi} & \xi \neq 0, \\ e^{-z} & \xi = 0, \end{cases}

for x \ge \mu, and the density is

f(x;\,\mu,\sigma,\xi) = \frac{1}{\sigma} (1 + \xi z)^{-1/\xi - 1}.

This is the limiting distribution of exceedances over a high threshold, so \mu is usually a fixed threshold rather than an estimated parameter. The support is x \ge \mu for \xi \ge 0 and \mu \le x \le \mu - \sigma/\xi for \xi < 0. At \xi = 0 the distribution is exponential with rate 1/\sigma, and at \xi > 0 with \mu = \sigma/\xi it is the Pareto distribution with \mu = 1/\xi.

The three cases are covered by one expression, so no branch on the sign or the value of \xi is needed. In particular the derivative with respect to \xi is exact at \xi = 0, which is the usual starting value when the shape is estimated.

The threshold itself belongs to the support: dgpd(mu, mu, sigma, xi) is 1/\sigma, as stats::dexp(0, rate) is rate. The VGAM, evd and extraDistr implementations return zero there instead.

Value

dgpd gives the density, pgpd gives the distribution function, qgpd gives the quantile function, and rgpd generates random deviates.

References

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer, doi:10.1007/978-1-4471-3675-0.

Pickands, J. (1975) Statistical inference using extreme order statistics. The Annals of Statistics, 3, 119-131.

See Also

gev, pareto, frechet

Examples

set.seed(123)
x <- rgpd(5, mu = 0, sigma = 1, xi = 0.3)
d <- dgpd(x, mu = 0, sigma = 1, xi = 0.3)
p <- pgpd(x, mu = 0, sigma = 1, xi = 0.3)
q <- qgpd(p, mu = 0, sigma = 1, xi = 0.3)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.