Generalised extreme value distribution

Density, distribution function, quantile function, and random generation for the generalised extreme value (GEV) distribution.

Usage

dgev(x, mu = 0, sigma = 1, xi = 0, log = FALSE)

pgev(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgev(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

rgev(n, mu = 0, sigma = 1, xi = 0)

Arguments

x, q: vector of quantiles
mu: location parameter
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

dgev gives the density, pgev gives the distribution function, qgev gives the quantile function, and rgev generates random deviates.

Details

dgev and pgev allow for automatic differentiation with RTMB.

With $z = (x - \mu) / \sigma$ the distribution function is $$F(x;\,\mu,\sigma,\xi) = \exp\bigl(-t(x)\bigr), \qquad t(x) = \begin{cases} (1 + \xi z)^{-1/\xi} & \xi \neq 0, \\ e^{-z} & \xi = 0, \end{cases}$$ and the density is $f(x) = t(x)^{\xi + 1} e^{-t(x)} / \sigma$.

The shape $\xi$ determines the tail and with it the support: for $\xi > 0$ (Frechet case) the distribution is heavy-tailed on $x > \mu - \sigma/\xi$, for $\xi < 0$ (Weibull case) it is bounded above by $\mu - \sigma/\xi$, and $\xi = 0$ is the Gumbel distribution on the whole real line.

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.

References

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values, Springer, doi:10.1007/978-1-4471-3675-0.

Jenkinson, A. F. (1955) The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Quarterly Journal of the Royal Meteorological Society, 81, 158-171.

Examples

set.seed(123)
x <- rgev(5, mu = 0, sigma = 1, xi = 0.2)
d <- dgev(x, mu = 0, sigma = 1, xi = 0.2)
p <- pgev(x, mu = 0, sigma = 1, xi = 0.2)
q <- qgev(p, mu = 0, sigma = 1, xi = 0.2)