frechet: Frechet distribution

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Frechet distribution

Description

Density, distribution function, quantile function, and random generation for the Frechet distribution.

Usage

dfrechet(x, mu = 0, sigma = 1, alpha = 1, log = FALSE)

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

qfrechet(p, mu = 0, sigma = 1, alpha = 1, lower.tail = TRUE, log.p = FALSE)

rfrechet(n, mu = 0, sigma = 1, alpha = 1)

Arguments

x, q	vector of quantiles
mu	location parameter, the lower end point of the support.
sigma	scale parameter, must be positive.
alpha	shape parameter, must be positive.
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

Details

dfrechet and pfrechet allow for automatic differentiation with RTMB.

With y = (x - \mu) / \sigma the density is

f(x;\,\mu,\sigma,\alpha) = \frac{\alpha}{\sigma} y^{-\alpha - 1} \exp(-y^{-\alpha}), \quad x > \mu,

and the distribution function is F(x) = \exp(-y^{-\alpha}).

The Frechet distribution is the heavy-tailed extreme value distribution: it is the generalised extreme value distribution with shape \xi = 1/\alpha > 0, reparameterised so that the shape enters as a tail index rather than as a reciprocal. All moments of order \alpha and above are infinite, so the mean exists only for \alpha > 1 and the variance only for \alpha > 2.

Value

dfrechet gives the density, pfrechet gives the distribution function, qfrechet gives the quantile function, and rfrechet generates random deviates.

References

Frechet, M. (1927) Sur la loi de probabilite de l'ecart maximum. Annales de la Societe Polonaise de Mathematique, 6, 93-116.

Kotz, S. and Nadarajah, S. (2000) Extreme Value Distributions: Theory and Applications, Imperial College Press, doi:10.1142/p191.

See Also

gev, gumbel, pareto

Examples

set.seed(123)
x <- rfrechet(5, mu = 0, sigma = 1, alpha = 3)
d <- dfrechet(x, mu = 0, sigma = 1, alpha = 3)
p <- pfrechet(x, mu = 0, sigma = 1, alpha = 3)
q <- qfrechet(p, mu = 0, sigma = 1, alpha = 3)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.