Frechet {extraDistr}

Frechet distribution

Description

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

Usage

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

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

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

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

Arguments

x, q

vector of quantiles.

lambda, sigma, mu

shape, scale, and location parameters. Scale and shape must be positive.

log, log.p

logical; if TRUE, probabilities p are given 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 observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(x) = \frac{\lambda}{\sigma} \left(\frac{x-\mu}{\sigma}\right)^{-1-\lambda} \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)

Cumulative distribution function

F(x) = \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)

Quantile function

F^{-1}(p) = \mu + \sigma -\log(p)^{-1/\lambda}

References

Bury, K. (1999). Statistical Distributions in Engineering. Cambridge University Press.

Examples

x <- rfrechet(1e5, 5, 2, 1.5)
xx <- seq(0, 1000, by = 0.1)
hist(x, 200, freq = FALSE)
lines(xx, dfrechet(xx, 5, 2, 1.5), col = "red") 
hist(pfrechet(x, 5, 2, 1.5))
plot(ecdf(x))
lines(xx, pfrechet(xx, 5, 2, 1.5), col = "red", lwd = 2)