revdbayes (version 1.5.5)

gp: The Generalised Pareto Distribution

Description

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

Usage

dgp(x, loc = 0, scale = 1, shape = 0, log = FALSE)
pgp(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
qgp(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
rgp(n, loc = 0, scale = 1, shape = 0)

Value

dgp gives the density function, pgp gives the distribution function, qgp gives the quantile function, and rgp generates random deviates.

Arguments

x, q

Numeric vectors of quantiles. All elements of x and q must be non-negative.

loc, scale, shape

Numeric vectors. Location, scale and shape parameters. All elements of scale must be positive.

log, log.p

A logical scalar; if TRUE, probabilities p are given as log(p).

lower.tail

A logical scalar. If TRUE (default), probabilities are \(P[X \leq x]\), otherwise, \(P[X > x]\).

p

A numeric vector of probabilities in [0,1].

n

Numeric scalar. The number of observations to be simulated. If length(n) > 1 then length(n) is taken to be the number required.

Details

The distribution function of a GP distribution with parameters location = \(\mu\), scale = \(\sigma (> 0)\) and shape = \(\xi\) is $$F(x) = 1 - [1 + \xi (x - \mu) / \sigma] ^ {-1/\xi}$$ for \(1 + \xi (x - \mu) / \sigma > 0\). If \(\xi = 0\) the distribution function is defined as the limit as \(\xi\) tends to zero. The support of the distribution depends on \(\xi\): it is \(x \geq \mu\) for \(\xi \geq 0\); and \(\mu \leq x \leq \mu - \sigma / \xi\) for \(\xi < 0\). Note that if \(\xi < -1\) the GP density function becomes infinite as \(x\) approaches \(\mu - \sigma/\xi\).

If lower.tail = TRUE then if p = 0 (p = 1) then the lower (upper) limit of the distribution is returned. The upper limit is Inf if shape is non-negative. Similarly, but reversed, if lower.tail = FALSE.

See https://en.wikipedia.org/wiki/Generalized_Pareto_distribution for further information.

References

Pickands, J. (1975) Statistical inference using extreme order statistics. Annals of Statistics, 3, 119-131. tools:::Rd_expr_doi("10.1214/aos/1176343003")

Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Chapter 4: tools:::Rd_expr_doi("10.1007/978-1-4471-3675-0_4")

Examples

dgp(0:4, scale = 0.5, shape = 0.8)
dgp(1:6, scale = 0.5, shape = -0.2, log = TRUE)
dgp(1, scale = 1, shape = c(-0.2, 0.4))

pgp(0:4, scale = 0.5, shape = 0.8)
pgp(1:6, scale = 0.5, shape = -0.2)
pgp(1, scale = c(1, 2), shape = c(-0.2, 0.4))
pgp(7, scale = 1, shape = c(-0.2, 0.4))

qgp((0:9)/10, scale = 0.5, shape = 0.8)
qgp(0.5, scale = c(0.5, 1), shape = c(-0.5, 0.5))

p <- (1:9)/10
pgp(qgp(p, scale = 2, shape = 0.8), scale = 2, shape = 0.8)

rgp(6, scale = 0.5, shape = 0.8)