#' @name KW
#'
#' @title
#' The Kumaraswamy Weibull Distribution
#'
#' @description
#' Density, distribution function, quantile function,
#' random generation and hazard function for the kumaraswamy weibull distribution with
#' parameters \code{alpha}, \code{theta}, \code{a} and \code{b}.
#'
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations.
#' @param alpha parameter one.
#' @param theta parameter two.
#' @param a parameter three.
#' @param b parameter four.
#' @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 kumaraswamy weibull with parameters \code{alpha}, \code{theta}, \code{a} and \code{b}
#' has density given by
#'
#' f(x)=a*b*alpha*theta*x^(theta-1)*exp(-alpha*x^theta)*(1-exp(-alpha*x^theta))^(a-1)*
#' (1-(1-exp(-aplha*x^theta))^a)^(b-1)
#'
#' for x>0.
#'
#' @return
#' \code{dKW} gives the density, \code{pKW} gives the distribution
#' function, \code{qKW} gives the quantile function, \code{rKW}
#' generates random deviates and \code{hKW} gives the hazard function.
#'
#' @export
#' @examples
#' ## The probability density function
#' curve(dKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5), from = 0, to = 3, ylim = c(0, 2.5), col = "red", las = 1, ylab = "The probability density function")
#'
#' ## The cumulative distribution and the Reliability function
#' par(mfrow = c(1, 2))
#' curve(pKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5), from = 0, to = 3, ylim = c(0, 1), col = "red", las = 1, ylab = "The cumulative distribution function")
#' curve(pKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5, lower.tail = FALSE), from = 0, to = 3, ylim = c(0, 1), col = "red", las = 1, ylab = "The Reliability function")
#'
#' ## The quantile function
#' p <- seq(0,0.99999, length.out=100)
#' plot(x=qKW(p, alpha = 3, theta = 0.8, a = 2, b = 1.5), y = p, xlab = "Quantile", las = 1, ylab = "Probability")
#' curve(pKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5), from = 0, add = TRUE, col = "red")
#'
#' ## The random function
#' hist(rKW(10000, alpha = 3, theta = 0.8, a = 2, b = 1.5), freq = FALSE, xlab = "x", las = 1, ylim = c(0, 2.5), main = "")
#' curve(dKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5), from = 0, add = TRUE, col = "red" )
#'
#' ## The Hazard function
#' curve(hKW(x, alpha = 3, theta = 0.8, a = 2, b = 1.5), from = 0, to = 2.1, ylim = c(0, 4), col = "red", ylab = "The Hazard function", las = 1)
dKW<-function(x,alpha,theta,a,b, 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(theta<=0))
stop(paste("theta must be positive", "\n", ""))
if (any(a<=0))
stop(paste("a must be positive", "\n", ""))
if (any(b<=0))
stop(paste("b must be positive", "\n", ""))
term <- -alpha*(x^theta)
loglik<- log(a*b*alpha*theta) + (theta-1)*log(x) + term +
(a-1)*log(1-exp(term)) + (b-1)*log(1-(1-exp(term))^a)
if (log == FALSE)
density<- exp(loglik)
else
density <- loglik
return(density)
}
#' @export
#' @rdname KW
pKW <- function(q,alpha,theta,a,b, 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(theta<=0))
stop(paste("theta must be positive", "\n", ""))
if (any(a<=0))
stop(paste("a must be positive", "\n", ""))
if (any(b<=0))
stop(paste("b must be positive", "\n", ""))
cdf <- 1-(1-(1-exp(-alpha*(q^theta)))^a)^b
if (lower.tail == TRUE)
cdf <- cdf
else cdf <- 1 - cdf
if (log.p == FALSE)
cdf <- cdf
else cdf <- log(cdf)
cdf
}
#' @export
#' @rdname KW
qKW <- function(p,alpha,theta,a,b, lower.tail = TRUE, log.p = FALSE){
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(theta<=0))
stop(paste("theta must be positive", "\n", ""))
if (any(a<=0))
stop(paste("a must be positive", "\n", ""))
if (any(b<=0))
stop(paste("b 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", ""))
q <- ((-1/alpha)*(log(1-(1-(1-p)^(1/b))^(1/a))))^(1/theta)
q
}
#' @export
#' @rdname KW
rKW <- function(n,alpha,theta,a,b){
if(any(n<=0))
stop(paste("n must be positive","\n",""))
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(theta<=0))
stop(paste("theta must be positive", "\n", ""))
if (any(a<=0))
stop(paste("a must be positive", "\n", ""))
if (any(b<=0))
stop(paste("b must be positive", "\n", ""))
n <- ceiling(n)
p <- runif(n)
r <- qKW(p,alpha,theta,a,b)
r
}
#' @export
#' @rdname KW
hKW<-function(x,alpha,theta,a,b){
if (any(x<0))
stop(paste("x must be positive", "\n", ""))
if (any(alpha<=0 ))
stop(paste("alpha must be positive", "\n", ""))
if (any(theta<=0))
stop(paste("theta must be positive", "\n", ""))
if (any(a<=0))
stop(paste("a must be positive", "\n", ""))
if (any(b<=0))
stop(paste("b must be positive", "\n", ""))
h <- dKW(x,alpha,theta,a,b, log = FALSE)/pKW(q=x,alpha,theta,a,b, lower.tail=FALSE, log.p = FALSE)
h
}
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