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)

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.

vector of probabilities.

number of observations.

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.

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

# NOT RUN {
dGPD(seq(1, 5), 0, 1, 1)
qGPD(pGPD(seq(1, 5), 0, 1, 1), 0, 1 ,1)
rGPD(5, 0, 1, 1)
# }