extraDistr (version 1.10.0)

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

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

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)