#' @name FWE
#'
#' @title
#' The Flexible Weibull Extension Distribution
#'
#' @description
#' Density, distribution function, quantile function,
#' random generation and hazard function for the flexible weibull extension distribution with
#' parameters \code{alpha} and \code{beta}.
#'
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations.
#' @param alpha parameter.
#' @param beta parameter.
#' @param log,log.p logical; if TRUE, probabilities p are given as log(p).
#' @param lower.tail logical; if TRUE (default), probabilities are
#' P[X <= x], otherwise, P[X > x].
#'
#' @details
#' The flexible weibull extension with parameters \code{alpha} and \code{beta}
#' has density given by
#'
#' f(x) = (alpha + (beta/x^2))*exp(alpha*x - beta/x)*exp(-exp(alpha*x-beta/x))
#'
#' for x>0.
#'
#' @return
#' \code{dFWE} gives the density, \code{pFWE} gives the distribution
#' function, \code{qFWE} gives the quantile function, \code{rFWE}
#' generates random deviates and \code{hFWE} gives the hazard function.
#'
#' @export
#' @examples
#' ## The probability density function
#' curve(dFWE(x, alpha = 0.75, beta = 0.5), from = 0, to = 3, ylim = c(0, 1.7), col = "red", las = 1, ylab = "The probability density function")
#'
#' ## The cumulative distribution and the Reliability function
#' par(mfrow = c(1, 2))
#' curve(pFWE(x, alpha = 0.75, beta = 0.5), from = 0, to = 3, col = "red", las = 1, ylab = "The cumulative distribution function")
#' curve(pFWE(x, alpha = 0.75, beta = 0.5, lower.tail = FALSE), from = 0, to = 3, col = "red", las = 1, ylab = "The Reliability function")
#'
#' ## The quantile function
#' p <- seq(from = 0, to = 0.99999, length.out = 100)
#' plot(x = qFWE(p,alpha = 0.75, beta = 0.5), y = p, xlab = "Quantile", las = 1, ylab = "Probability")
#' curve(pFWE(x, alpha = 0.75, beta = 0.5), from = 0, add = TRUE, col = "red")
#'
#' ## The random function
#' hist(rFWE(n = 1000, alpha = 2, beta = 0.5), freq = FALSE, xlab = "x", ylim = c(0, 2), las = 1, main = "")
#' curve(dFWE(x, alpha = 2, beta = 0.5), from = 0, to = 3, add = TRUE, , col = "red")
#'
#' ## The Hazard function
#' curve(hFWE(x, alpha = 0.75, beta = 0.5), from = 0, to = 2, ylim = c(0, 2.5), col = "red", ylab = "The Hazard function", las = 1)
#'
dFWE<-function(x,alpha,beta,log = FALSE){
if (any(x<0))
stop(paste("x must be positive", "\n", ""))
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(beta<=0))
stop(paste("beta must be positive", "\n", ""))
loglik<- log(alpha + (beta/x^2)) + (alpha*x) - (beta/x) -
exp(alpha*x - (beta/x))
if (log == FALSE)
density<- exp(loglik)
else
density <- loglik
return(density)
}
#' @export
#' @rdname FWE
pFWE <- function(q,alpha,beta, lower.tail=TRUE, log.p = FALSE){
if (any(q<0))
stop(paste("q must be positive", "\n", ""))
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(beta<=0))
stop(paste("beta must be positive", "\n", ""))
cdf <- 1- exp(-exp(alpha*q - beta/q))
if (lower.tail == TRUE)
cdf <- cdf
else cdf <- 1 - cdf
if (log.p == FALSE)
cdf <- cdf
else cdf <- log(cdf)
cdf
}
#' @export
#' @rdname FWE
qFWE <- function(p, alpha, beta, lower.tail = TRUE, log.p = FALSE) {
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(beta<=0))
stop(paste("beta must be positive", "\n", ""))
if (log.p == TRUE)
p <- exp(p)
else p <- p
if (lower.tail == TRUE)
p <- p
else p <- 1 - p
if (any(p < 0) | any(p > 1))
stop(paste("p must be between 0 and 1", "\n", ""))
fda <- function(x,alpha, beta){
1- exp(-exp(alpha*x - beta/x))
}
fda1 <- function(x, alpha, beta, p) {fda(x, alpha, beta) - p}
r_de_la_funcion <- function(alpha, beta, p) {
uniroot(fda1, interval=c(0,1e+06), alpha, beta, p)$root
}
r_de_la_funcion <- Vectorize(r_de_la_funcion)
q <- r_de_la_funcion(alpha, beta, p)
q
}
#' @export
#' @rdname FWE
rFWE <- function(n,alpha,beta){
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(beta<=0))
stop(paste("beta must be positive", "\n", ""))
n <- ceiling(n)
p <- runif(n)
r <- qFWE(p, alpha,beta)
r
}
#' @export
#' @rdname FWE
hFWE<-function(x,alpha,beta){
if (any(x<0))
stop(paste("x must be positive", "\n", ""))
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(beta<=0))
stop(paste("beta must be positive", "\n", ""))
h <- dFWE(x,alpha,beta, log = FALSE)/pFWE(q=x,alpha,beta, lower.tail=FALSE, log.p = FALSE)
h
}
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