The Generalised Weibull Distribution

The Generalised Weibull Distribution

The Generalised Weibull distribution was proposed by [1]. This distribution is defined in terms of its quantile function: \[ Q(u;\sigma,\alpha,\lambda) = \sigma\left[\dfrac{1-(1-u)^\lambda}{\lambda}\right]^\alpha, \,\,\,\,\, \text{if} \,\,\, \lambda\neq 0,\\ Q(u;\sigma,\alpha,\lambda) = \sigma\left[-\log(1-u)\right]^\alpha, \,\,\,\,\, \text{if} \,\,\, \lambda = 0. \] where \(\alpha>0\), \(\sigma>0\), and \(\lambda \in{\mathbb R}\). The cumulative distribution function is given by \[ F(t;\sigma,\alpha,\lambda) = 1 - ( 1-\lambda(t/\sigma)^{1/\alpha})^{1/\lambda}. \] The probability density function is: \[ f(t;\sigma,\alpha,\lambda) = \frac{\left(\frac{t}{\sigma }\right)^{\frac{1}{\alpha } - 1} \left(1-\lambda \left(\frac{t}{\sigma }\right)^{\frac{1}{\alpha }}\right)^{\frac{1}{\lambda }-1}}{\alpha \sigma}, \] while the hazard function is \[ h(t;\sigma,\alpha,\lambda) = \dfrac{(t/\sigma)^{\frac{1}{\alpha}-1}}{\alpha\sigma\left( 1-\lambda(t/\sigma)^{1/\alpha} \right)}. \] The support of the generalised Weibull distribution is \((0,\infty)\) for \(\lambda\leq 0\) and \((0,\sigma/\lambda^\alpha)\) for \(\lambda>0\).

The following R code contains an implementation of the cumulative distribution function, quantile function, hazard function, and cumulative hazard function of the Generalised Weibull distribution.

References

1. A Generalization of the Weibull Distribution with Application to the Analysis of Survival Data

rm(list=ls())
# t         : vector of quantiles.
# p         : vector of probabilities.
# sigma     : scale parameter
# lambda    : shape parameter
# alpha     : shape parameter

# Quantile function
qgenweibull<- function(p,sigma,alpha,lambda){
  if(lambda==0) quant <- sigma*( -log(1-p) )^alpha
  else quant <-  sigma*( (1-(1-p)^lambda)/lambda )^alpha
  return(quant)
}  

# Cumulative distribution function
pgenweibull<- function(t,sigma,alpha,lambda,log.p=FALSE){
  log.cdf <- log( 1 - ( 1 - lambda*( t/sigma )^(1/alpha)  )^(1/lambda) )
    ifelse(log.p, return(log.cdf), return(exp(log.cdf)))
}  

# Probability density function
dgenweibull<- function(t,sigma,alpha,lambda,log.p=FALSE){
  tsig <- t/sigma
  log.pdf <-   (1/alpha -1)*log(tsig) + (1/lambda - 1)*log( 1 - lambda*tsig^(1/alpha)) -
    log(alpha) - log(sigma)
  ifelse(log.p, return(log.pdf), return(exp(log.pdf)))
} 

# Hazard function
hgenweibull<- function(t,sigma,alpha,lambda,log.p=FALSE){
  log.h <- (1/alpha-1)*log(t/sigma) - log(alpha) - log(sigma) - log( 1 - lambda*( t/sigma )^(1/alpha) )
  ifelse(log.p, return(log.h), return(exp(log.h)))
}  

# Cumulative hazard function
CHgenweibull <- function(t,sigma,alpha,lambda,log.p=FALSE){
  log.CH <- log( 1 - lambda*( t/sigma )^(1/alpha) )/lambda 
  ifelse(log.p, return(log.CH), return(exp(log.CH)))
} 


# Examples of the shape of the hazard function for different values of the parameters
h1 <- Vectorize(function(t) hgenweibull(t,1,1,0))
h2 <- Vectorize(function(t) hgenweibull(t,1,1.5,0))
h3 <- Vectorize(function(t) hgenweibull(t,1.9,1.75,0.55))
h4 <- Vectorize(function(t) hgenweibull(t,0.9,1,0.15))
h5 <- Vectorize(function(t) hgenweibull(t,0.25,0.8,-1 ))

#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(h2,0,4,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)