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#' Frechet distribution
#'
#' Density, distribution function, quantile function and random generation
#' for the Frechet distribution.
#'
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations. If \code{length(n) > 1},
#' the length is taken to be the number required.
#' @param lambda,sigma,mu shape, scale, and location parameters.
#' Scale and shape must be positive.
#' @param log,log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are \eqn{P[X \le x]}
#' otherwise, \eqn{P[X > x]}.
#'
#' @details
#'
#' Probability density function
#' \deqn{
#' f(x) = \frac{\lambda}{\sigma} \left(\frac{x-\mu}{\sigma}\right)^{-1-\lambda} \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)
#' }{
#' f(x) = \lambda/\sigma * ((x-\mu)/\sigma)^(-1-\lambda) * exp(-((x-\mu)/\sigma)^-\lambda)
#' }
#'
#' Cumulative distribution function
#' \deqn{
#' F(x) = \exp\left(-\left(\frac{x-\mu}{\sigma}\right)^{-\lambda}\right)
#' }{
#' F(x) = exp(-((x-\mu)/\sigma)^-\lambda)
#' }
#'
#' Quantile function
#' \deqn{
#' F^{-1}(p) = \mu + \sigma -\log(p)^{-1/\lambda}
#' }{
#' 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)
#'
#' @name Frechet
#' @aliases Frechet
#' @aliases dfrechet
#'
#' @keywords distribution
#' @concept Univariate
#' @concept Continuous
#'
#' @export
dfrechet <- function(x, lambda = 1, mu = 0, sigma = 1, log = FALSE) {
cpp_dfrechet(x, lambda, mu, sigma, log[1L])
}
#' @rdname Frechet
#' @export
pfrechet <- function(q, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) {
cpp_pfrechet(q, lambda, mu, sigma, lower.tail[1L], log.p[1L])
}
#' @rdname Frechet
#' @export
qfrechet <- function(p, lambda = 1, mu = 0, sigma = 1, lower.tail = TRUE, log.p = FALSE) {
cpp_qfrechet(p, lambda, mu, sigma, lower.tail[1L], log.p[1L])
}
#' @rdname Frechet
#' @export
rfrechet <- function(n, lambda = 1, mu = 0, sigma = 1) {
if (length(n) > 1) n <- length(n)
cpp_rfrechet(n, lambda, mu, sigma)
}
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