Fréchet Distribution

The Fréchet distribution, also known as the Type II extreme value distribution, is a continuous probability distribution for the maximum of a sequence of independent random variables. It has a lower bound and a heavy right tail, and is parameterized by location, scale, and shape.

Usage

dfrechet(x, loc = 0, scale = 1, shape = 1, log = FALSE)

pfrechet(q, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

qfrechet(p, loc = 0, scale = 1, shape = 1, lower.tail = TRUE, log.p = FALSE)

rfrechet(n, loc = 0, scale = 1, shape = 1)

Arguments

x, q: vector of quantiles.
loc, scale, shape: location, scale and shape parameters (can be given as vectors).
log, log.p: logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.
lower.tail: logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
p: vector of probabilities.
n: number of observations.

Value

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

Details

Density function, distribution function, quantile function and random generation for the Frechet distribution with location, scale and shape parameters.

The Frechet distribution function with parameters $`loc` = a$, $`scale` = b$ and $`shape` = s$ is $$G(z) = \exp\left\{-\left(\frac{z-a}{b}\right)^{-s}\right\}$$ for $z > a$ and zero otherwise, where $b > 0$ and $s > 0$.

Note

Based on code by Alec Stephenson previously published in the evd package, adapted to conform to package standards.

Examples


dfrechet(2:4, 1, 0.5, 0.8)
#> [1] 0.25871959 0.09487423 0.05010381
pfrechet(2:4, 1, 0.5, 0.8)
#> [1] 0.5630712 0.7190122 0.7878127
qfrechet(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
#> [1] 10.329571  5.260165  3.813966  3.157788
rfrechet(6, 1, 0.5, 0.8)
#> [1] 1.180762 1.836321 2.017466 9.541024 5.224901 1.860130
p <- (1:9)/10
pfrechet(qfrechet(p, 1, 2, 0.8), 1, 2, 0.8)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9