GPD: The Generalized Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the generalized Pareto distribution with location, scale and shape parameters.

Usage

dGPD(x, loc = 0, scale = 1, shape = 0, log = FALSE)
pGPD(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
qGPD(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)
rGPD(n, loc = 0, scale = 1, shape = 0)

Value

dGPD gives the density, pGPD gives the distribution function, qGPD gives the quantile function, and

rGPD generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

Arguments

x, q: vector of quantiles.
loc: location parameter.
scale: scale parameter.
shape: shape parameter.
log, log.p: logical; if TRUE, probabilities $p$ are given as $log(p)$, default: FALSE.
lower.tail: logical; if TRUE, probabilities are $P[X \le x]$ otherwise, $P[X > x]$, default: TRUE.
p: vector of probabilities.
n: number of observations.

Details

The generalized Pareto distribution function with location parameter $\mu$, scale parameter $\sigma$ and shape parameter $\xi$ has density given by $$f(x)=1/\sigma (1 + \xi z)^-(1/\xi + 1)$$ for $x\ge \mu$ and $\xi> 0$, or $\mu-\sigma/\xi \ge x\ge \mu$ and $\xi< 0$, where $z=(x-\mu)/\sigma$. In the case where $\xi= 0$, the density is equal to $f(x)=1/\sigma e^-z$ for $x\ge \mu$. The cumulative distribution function is $$F(x)=1-(1+\xi z)^(-1/\xi)$$ for $x\ge \mu$ and $\xi> 0$, or $\mu-\sigma/\xi \ge x\ge \mu$ and $\xi< 0$, with $z$ as stated above. If $\xi= 0$ the CDF has form $F(x)=1-e^-z$.

See https://en.wikipedia.org/wiki/Generalized_Pareto_distribution for more details.

Examples

Run this code

dGPD(seq(1, 5), 0, 1, 1)
qGPD(pGPD(seq(1, 5), 0, 1, 1), 0, 1 ,1)
rGPD(5, 0, 1, 1)