Generalised Pareto distribution

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

Value

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

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.

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.

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)