gp {revdbayes}

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)

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.

Value

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

References

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

Coles, S. G. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, London. Chapter 4: 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)