#' @name RW
#'
#' @title
#' The Reflected Weibull Distribution
#'
#' @description
#' Density, distribution function, quantile function,
#' random generation and hazard function for the reflected weibull distribution with
#' parameters \code{mu} and \code{sigma}.
#'
#' @param x,q vector of quantiles.
#' @param p vector of probabilities.
#' @param n number of observations.
#' @param mu parameter one.
#' @param sigma parameter two.
#' @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 reflected weibull distribution with parameters \code{mu} and
#' \code{sigma} has density given by
#'
#' f(x) = mu*sigma*(-x)^(sigma-1)*exp(-mu*(-x)^mu)
#'
#' for - inf < x < 0.
#'
#' @return
#' \code{dRW} gives the density, \code{pRW} gives the distribution
#' function, \code{qRW} gives the quantile function, \code{rRW}
#' generates random deviates and \code{hRW} gives the hazard function.
#'
#' @export
#' @examples
#' ## The probability density function
#' curve(dRW(x, mu = 1, sigma = 1), from = -5, to = 0, ylim = c(0, 1), col = "red", las = 1, ylab = "The probability density function")
#'
#' ## The cumulative distribution and the Reliability function
#' par(mfrow = c(1, 2))
#' curve(pRW(x, mu = 1, sigma = 1), from = -5, to = 0, ylim = c(0, 1), col = "red", las = 1, ylab ="The cumulative distribution function")
#' curve(pRW(x, mu = 1, sigma = 1, lower.tail = FALSE), from = -5, to = 0, ylim = c(0, 1), col = "red", las = 1, ylab = "The Reliability function")
#'
#' ## The quantile function
#' p <- seq(from = 0, to = 0.99999, length.out = 100)
#' plot(x=qRW(p=p,mu = 1, sigma = 1), y=p, xlab="Quantile", las=1, ylab="Probability")
#' curve(pRW(x, mu = 1, sigma = 1), from = -5, add = TRUE, col = "red")
#'
#' ## The random function
#' hist(rRW(n = 10000, mu = 1, sigma = 1), freq = FALSE,xlab = "x", las = 1, main = "")
#' curve(dRW(x, mu = 1, sigma = 1), from = -5, to = 0, add = TRUE, col = "red")
#'
#' ## The Hazard function
#' curve(hRW(x, mu = 1, sigma = 1), from = -5, to = 0, ylim = c(0, 1), col = "red", ylab = "The hazard function", las = 1)
RW <- function (mu.link="log", sigma.link="log")
{
mstats <- checklink("mu.link" , "Reflected Weibull", substitute(mu.link), c("log", "own"))
dstats <- checklink("sigma.link", "Reflected Weibull", substitute(sigma.link), c("log", "own"))
structure(list(family = c("RW", "Reflected Weibull"),
parameters = list(mu=TRUE, sigma=TRUE),
nopar = 2,
type = "Continuous",
mu.link = as.character(substitute(mu.link)),
sigma.link = as.character(substitute(sigma.link)),
mu.linkfun = mstats$linkfun,
sigma.linkfun = dstats$linkfun,
mu.linkinv = mstats$linkinv,
sigma.linkinv = dstats$linkinv,
mu.dr = mstats$mu.eta,
sigma.dr = dstats$mu.eta,
dldm = function(y,mu,sigma) ((1/mu) -((-y)^sigma) ),
d2ldm2 = function(mu) (-1/mu^2),
dldd = function(y,mu,sigma) ((1/sigma)+log(-y)-mu*(-((-y)^sigma)*log(-y))),
d2ldd2 = function(y,mu,sigma) {
dldd = function(y,mu,sigma) ((-1/(sigma^2))-mu*((-y)^sigma)*(log(-y))^2)
ans <- dldd(y,mu,sigma)
ans <- -ans^2
},
d2ldmdd = function(y,mu,sigma) -(((-y)^sigma)*log(-y))^2,
G.dev.incr = function(y,mu,sigma,...) -2*dRW(y, mu, sigma, log=TRUE),
rqres = expression(rqres(pfun="pRW", type="Continuous", y=y, mu=mu, sigma=sigma)),
mu.initial = expression( mu <- rep(0.5, length(y)) ),
sigma.initial = expression( sigma <- rep(0.5, length(y)) ),
mu.valid = function(mu) all(mu > 0) ,
sigma.valid = function(sigma) all(sigma > 0),
y.valid = function(y) all(y < 0)
),
class = c("gamlss.family","family"))
}
#' @export
#' @rdname RW
dRW<-function(x,mu,sigma, log=FALSE){
if (any(x>0))
stop(paste("x must be negative", "\n", ""))
if (any(mu <= 0 ))
stop(paste("mu must be positive", "\n", ""))
if (any(sigma<=0))
stop(paste("sigma must be positive", "\n", ""))
loglik<- log(mu) + log(sigma) + (sigma-1)*log(-x) -
mu*((-x)^sigma)
if (log == FALSE)
density<- exp(loglik)
else
density <- loglik
return(density)
}
#' @export
#' @rdname RW
pRW <- function(q,mu,sigma, lower.tail=TRUE, log.p = FALSE){
# if (any(q<0))
# stop(paste("q must be positive", "\n", ""))
if (any(mu <= 0 ))
stop(paste("mu must be positive", "\n", ""))
if (any(sigma<=0))
stop(paste("sigma must be positive", "\n", ""))
cdf <- exp(-mu*(-q)^sigma)
if (lower.tail == TRUE)
cdf <- cdf
else cdf <- 1 - cdf
if (log.p == FALSE)
cdf <- cdf
else cdf <- log(cdf)
cdf
}
#' @export
#' @rdname RW
qRW <- function(p,mu,sigma, lower.tail = TRUE, log.p = FALSE){
if (any(mu <= 0 ))
stop(paste("mu must be positive", "\n", ""))
if (any(sigma<=0))
stop(paste("sigma 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/mu)*log(p))^(1/sigma)}
q
}
#' @export
#' @rdname RW
rRW <- function(n,mu,sigma){
if(any(n<=0))
stop(paste("n must be positive","\n",""))
if (any(mu<=0 ))
stop(paste("mu must be positive", "\n", ""))
if (any(sigma<=0))
stop(paste("sigma must be positive", "\n", ""))
n <- ceiling(n)
p <- runif(n)
r <- qRW(p,mu,sigma)
r
}
#' @export
#' @rdname RW
hRW<-function(x,mu,sigma){
if (any(x>0))
stop(paste("x must be negative", "\n", ""))
if (any(mu <= 0 ))
stop(paste("mu must be positive", "\n", ""))
if (any(sigma<=0))
stop(paste("sigma must be positive", "\n", ""))
h <- dRW(x,mu,sigma, log = FALSE)/pRW(q=x,mu,sigma, lower.tail=FALSE, log.p = FALSE)
h
}
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