Frechet distribution

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

Value

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

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 $\alpha > 2$.

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.

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)