Bayesian survival and hazard estimates for Weibull regression with censored data using modified Jeffreys prior

In this study, firstly, consideration is given to the traditional maximum likelihood estimator and the Bayesian estimator by employing Jeffreys prior and Extension of Jeffreys prior information on the Weibull distribution with a given shape under right censored data. We have formulated equations for the scale parameter, the survival function and the hazard functionunder Bayesian with extension of Jeffreys prior. Next we consider both the scale and shape parameters to be unknown under censored data. It is observed that the estimate of the shape parameter under the maximum likelihood method cannot be obtained in closed form, but can be solved by the application of numerical methods. With the application of the Bayesian estimates for the parameters, the survival function and hazard function, we realised that the posterior distribution from which Bayesian inference is drawn cannot be obtained analytically. Due to this, we have employed Lindley’s approximation technique and then compared it to the maximum likelihood approach. We then incorporate covariates into the Weibull model. Under this regression model with regards to Bayesian, the usual method was not possible. Thus we develop an approach to accommodate the covariate terms in the Jeffreys and Modified of Jeffreys prior by employingGauss quadrature method. Subsequently, we use Markov Chain Monte Carlo (MCMC) method in the Bayesian estimator of the Weibull distributionand Weibull regression model with shape unknown. For the Weibull model with right censoring and unknown shape, the full conditional distribution for the scale and shape parameters are obtained via Gibbs sampling and Metropolis-Hastings algorithm from which the survival function and hazard function are estimated. For Weibull regression model of both Jeffreys priors with covariates, importance sampling technique has been employed. Mean squared error (MSE) and absolute bias are obtained and used to compare the Bayesian and the maximum likelihood estimation through simulation studies. Lastly, we use real data to assess the performance of the developed models based on Gauss quadrature and Markov Chain Monte Carlo (MCMC) methods together with the maximum likelihood approach. The comparisons are done by using standard error and the confidence interval for maximum likelihood method and credible interval for the Bayesian method