Reliability estimation under type-II censored data from the generalized Bilal distribution

Abstract

The main object of this article is the estimation of the unknown population parameters and the reliability function for the generalized Bilal model under type-II censored data. Both maximum likelihood and Bayesian estimates are considered. In the Bayesian framework, although we have discussed mainly the squared error loss function, any other loss function can easily be considered. Gibb's sampling procedure is used to draw Markov Chain Monte Carlo (MCMC) samples, which have been used to compute the Bayes estimates and also to construct their corresponding credible intervals with the help of two different importance sampling techniques. A simulation study is carried out to examine the accuracy of the resulting Bayesian estimates and compare them with their corresponding maximum likelihood estimates. Application to a real data set is considered for the sake of illustration.

Discrete Bilal distribution with right-censored data

“…L <-sum(log(like)) if (is.na(L)==TRUE) {return(-Inf)} else {return(L)} } # Obtaining the ML estimates mle <-c() mle <-maxLik(logLik=log.f,start=c(0.08,0.6)) summary(mle) betaDB <-mle$estimate [1] etaDB <-mle$estimate [2] s <-vcov(mle) # The 95% confidence intervals llimDB <-round(betaDB -qnorm(0.975) * sqrt(s [1,1]),4) ulimDB <-round(betaDB + qnorm(0.975) * sqrt(s[1,1]),4) llimDBe <-round(etaDB -qnorm(0.975) * sqrt(s [2,2]),4) ulimDBe <-round(etaDB + qnorm(0.975) * sqrt(s [2,2]),4) cat("n = ",n,"\n") cat("Beta = ",betaDB, "95%CI: (",llimDB,",",ulimDB, ") \n") cat("Eta = ",etaDB, "95%CI: (",llimDBe,",",ulimDBe, ") \n") # Calculating AIC, BIC and AICC aic <-AIC(mle) bic <-AIC(mle,k = log(n)) aicc <-aic + (2*K^2+2*K)/(n-K-1) cat("AIC = ",aic,", BIC = ",bic,", AICC = ",aicc,"\n") This is the R code for the Bayesian model for survival data with a cure fraction based on the DB distribution, as presented in subsection 2. <-dgamma(beta,0.001,0.001)*dbeta(eta,1,1) log.prior <-log(prior) L <-log.like + log.prior if (is.na(L)==TRUE) {return(-Inf)} else {return(L)} } # Obtaining the MCMC estimates posterior <-MCMCmetrop1R(log.post,theta.init=c(beta=0.05,eta=0.6), burnin=10000, mcmc=1000000, thin=200, logfun=T, t=t, d=d, verbose=100000, tune = 1) varnames(posterior) <-c("beta","eta") summary(posterior)…”

Discrete Bilal Distribution in the Presence of Right-Censored Data and a Cure Fraction

“…# Obtaining the ML estimates mle <-c() mle <-maxLik(logLik=log.f,start=c(0.08,0.6)) summary(mle) betaDB <-mle$estimate [1] etaDB <-mle$estimate [2] s <-vcov(mle) # The 95% confidence intervals llimDB <-round(betaDB -qnorm(0.975) * sqrt(s[1,1]),4) 1186 DISCRETE BILAL DISTRIBUTION WITH RIGHT-CENSORED DATA ulimDB <-round(betaDB + qnorm(0.975) * sqrt(s[1,1]),4) llimDBe <-round(etaDB -qnorm(0.975) * sqrt(s[2,2]),4) ulimDBe <-round(etaDB + qnorm(0.975) * sqrt(s[2,2]),4) cat("n = ",n,"\n") cat("Beta = ",betaDB, "95%CI: (",llimDB,",",ulimDB, ") \n") cat("Eta = ",etaDB, "95%CI: (",llimDBe,",",ulimDBe, ") \n") # Calculating AIC, BIC and AICC aic <-AIC(mle) bic <-AIC(mle,k = log(n)) aicc <-aic + (2 * Kˆ2+2 * K)/(n-K-1) cat("AIC = ",aic,", BIC = ",bic,", AICC = ",aicc,"\n") This is the R code for the Bayesian model for survival data with a cure fraction based on the DB distribution, as presented in subsection 2.4: <-dgamma(beta,0.001,0.001) * dbeta(eta,1,1) log.prior <-log(prior) L <-log.like + log.prior if (is.na(L)==TRUE) {return(-Inf)} else {return(L)} } # Obtaining the MCMC estimates posterior <-MCMCmetrop1R(log.post,theta.init=c(beta=0.05,eta=0.6), burnin=10000, mcmc=1000000, thin=200, logfun=T, t=t, d=d, verbose=100000, tune = 1) varnames(posterior) <-c("beta","eta") summary(posterior) # Obtaining the HPD intervals HPDinterval(posterior, prob = 0.95) # Geweke z scores geweke.diag(posterior)…”

Estimation for Entropy and Parameters of Generalized Bilal Distribution under Adaptive Type II Progressive Hybrid Censoring Scheme

“…A comprehensive mathematical treatment of the GB distribution was provided, and the maximum likelihood estimations of unknown parameters were derived under the complete sample. Abd-Elrahman [ 21 ] provided the MLEs and Bayesian estimations of the unknown parameters and the reliability function based on a Type-II censored sample. Since the failure rate function of GB distribution has an upside-down bathtub shape, and it can also be monotonically decreasing or monotonically increasing at some selected values of the shape parameters , the GB model is very useful in survival analysis and reliability studies.…”