Accelerated life testing using the Eyring model for the Weibull and Birnbaum- Saunders distributions

Abstract

In this thesis, we present a novel approach to new Bayesian dual-stress accelerated life testing models. The generalised Eyring model, with one thermal stressor and one non-thermal stressor, is utilised as the time transformation function. The new models use the Weibull and Birnbaum-Saunders distributions as the life distributions. General likelihood formulations are given for the models, which can accommodate uncensored, type-I censored and type-II censored data. Variations for the generalised Eyring-Weibull model are presented via different choices of prior distributions, which include uniform, gamma, and log-normal priors. For the generalised Eyring-Birnbaum-Saunders model, gamma priors are imposed on the model parameters. The full conditional and joint posterior distributions for the models are presented. The models have mathematically intractable posterior distributions, which means that Markov chain Monte Carlo methods need to be employed to generate posterior samples for inference. The log-concavity of the generalised Eyring-Weibull models is assessed to determine which Markov chain Monte Carlo methods are appropriate to use. The new models are applied to a real data set, where temperature and relative humidity are the accelerated stressors. The sensitivity of the models is investigated by specifying various values for the hyperparameters. The models are implemented in OpenBUGS to generate posterior samples. The convergence of the Markov chains is monitored using trace plots and the Brooks-Gelman-Rubin approach. The Monte Carlo error is used to determine if an adequate number of samples have been generated by the Markov chains. The fit of the models is assessed via the deviance information criterion. Inferential results, such as summary statistics, marginal posterior distributions and the predictive reliability, are presented and compared between the models. It is found that both models are sensitive to the spescific choice of subjective priors, specifically when the prior variance is small. It is recommended that flat priors should ideally be used if no prior information is available. The use of Bayes factors for model selection in accelerated life testing is also explored. Due to the mathematically intractable posterior distributions of the new models, the marginal likelihood for Bayes factors must be estimated. The focus is on methods that can approximate the marginal likelihood from the samples generated by a Markov chain Monte Carlo algorithm. These methods include a simple Monte Carlo estimator, the harmonic mean estimator, the Laplace-Metropolis estimator, and a posterior predictive density estimate for posterior Bayes factors. The new models are applied to another real data set and implemented in OpenBUGS to generate posterior samples. The Bayes factors and posterior model probabilities are calculated using different estimators. Model selection is carried out by comparing the Bayes factors and the deviance information criterion.