The Exponentiated Weibull distribution

The Exponentiated Weibull distribution

The Exponentiated Weibull distribution is a generalisation of the Weibull distribution which is obtained by exponentiating the Weibull cumulative distribution function. This simple transformation adds a second shape parameter that, interestingly, induces a lot of flexibility on the hazard function. The hazard function of the Exponentiated Weibull distribution can capture the basic shapes: constant, increasing, decreasing, bathtub, and unimodal, making it appealing for survival models.

The probability density function and cumulative distribution function of the Exponentiated Weibull distribution are respectively given by: \[f_{EW}(t) = \alpha \dfrac{\kappa}{\lambda} \left(\dfrac{t}{\lambda}\right)^{\kappa-1} \left[1-\exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}\right]^{\alpha-1} \exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}, \\ F_{EW}(t) = \left[1-\exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}\right]^{\alpha},\] where \(t>0\), \(\alpha>0\), \(\lambda>0\), and \(\kappa>0\). The parameter \(\lambda\) is a scale parameter, \(\kappa\) is a shape parameter, and \(\alpha\) is the power (shape) parameter. The following R code contains an implementation of the probability density function, cumulative distribution function, quantile function, random number generation, hazard function, and cumulative hazard function of the Exponentiated Weibull distribution using the commands already implemented in R for the Weibull distribution.

References.

1. A comparison of the generalized gamma and exponentiated Weibull distributions

2. Log-location-scale-log-concave distributions for survival and reliability analysis

rm(list=ls())
# t, q      : vector of quantiles.
# p         : vector of probabilities.
# n         : number of observations. 
# lambda : scale parameter
# kappa  : shape parameter
# alpha  : power (shape) parameter

# Cumulative distribution function
pexpweibull<- function(t,lambda,kappa,alpha,log.p=FALSE){
  log.cdf <- alpha*pweibull(t,scale=lambda,shape=kappa,log.p=TRUE)
  ifelse(log.p, return(log.cdf), return(exp(log.cdf)))
}  

# Probability density function
dexpweibull<- function(t,lambda,kappa,alpha,log=FALSE){
  log.pdf <-  log(alpha) + (alpha-1)*pweibull(t,scale=lambda,shape=kappa,log=TRUE) + 
                dweibull(t,scale=lambda,shape=kappa,log=TRUE)
  ifelse(log, return(log.pdf), return(exp(log.pdf)))
}

# Quantile function
qexpweibull<- function(p,lambda,kappa,alpha){
  quant <-  qweibull(p^(1/alpha),scale=lambda,shape=kappa)
   return(quant)
}  

# Random number generation
rexpweibull<- function(n,lambda,kappa,alpha){
  u = runif(n)
  sim <-  qweibull(u^(1/alpha),scale=lambda,shape=kappa)
  return(sim)
} 

# Hazard function
hexpweibull <- function(t,lambda,kappa,alpha,log=FALSE){
  log.pdf <-  log(alpha) + (alpha-1)*pweibull(t,scale=lambda,shape=kappa,log.p=TRUE) + 
    dweibull(t,scale=lambda,shape=kappa,log=TRUE)
  cdf <- exp(alpha*pweibull(t,scale=lambda,shape=kappa,log.p=TRUE) )
  log.h <- log.pdf - log(1-cdf)
ifelse(log, return(log.h), return(exp(log.h)))
}                                                                                      

# Cumulative hazard function
CHexpweibull <- function(t,lambda,kappa,alpha,log.p=FALSE){
  cdf <- exp(alpha*pweibull(t,scale=lambda,shape=kappa,log.p=TRUE) )
  return(-log(1-cdf))
} 

# Examples of the shape of the hazard function for different values of the parameters
h1 <- Vectorize(function(t) hexpweibull(t,1,1,1))
h2 <- Vectorize(function(t) hexpweibull(t,1,1.5,1))
h3 <- Vectorize(function(t) hexpweibull(t,0.5,0.5,0.2))
h4 <- Vectorize(function(t) hexpweibull(t,3,4,0.005))
h5 <- Vectorize(function(t) hexpweibull(t,0.000005 ,0.15,110 ))

#par(mfrow = c(3,2))
curve(h1,0,5,ylim=c(0,5),col="black",lwd=2,n=1000,xlab="t",ylab="Hazard function",cex.axis=1.5,cex.lab=1.5)

curve(h2,0,5,ylim=c(0,5),col="blue",lwd=2,n=1000,xlab="t",ylab="Hazard function",cex.axis=1.5,cex.lab=1.5)

curve(h3,0,5,ylim=c(0,5),col="red",lwd=2,n=1000,xlab="t",ylab="Hazard function",cex.axis=1.5,cex.lab=1.5)

curve(h4,0,5,ylim=c(0,5),col="green",lwd=2,n=1000,xlab="t",ylab="Hazard function",cex.axis=1.5,cex.lab=1.5)

curve(h5,0,5,ylim=c(0,5),col="purple",lwd=2,n=1000,xlab="t",ylab="Hazard function",cex.axis=1.5,cex.lab=1.5)