Bayesian accelerated life tests for the Weibull distribution under non-informative priors

In a competitive world where products are designed to last for long periods of time, obtaining time-to-failure data is both difficult and costly. Hence for products with high reliability, accelerated life testing is required to obtain relevant life-data quickly. This is done by placing the products under higher-than-use stress levels, thereby causing the products to fail prematurely. Part of the analysis of accelerated life-data requires a life distribution that describes the lifetime of a product at a given stress level and a life-stress relationship – which is some function that describes the way in which the life distribution changes across different stress levels. In this thesis it is assumed that the underlying life distribution is the wellknown Weibull distribution, with shape parameter constant over all stress levels and scale parameter as a log-linear function of stress. The primary objective of this thesis is to obtain estimates from Bayesian analysis, and this thesis considers five types of non-informative prior distributions: Jeffreys’ prior, reference priors, maximal data information prior, uniform prior and probability matching priors. Since the associated posterior distribution under all the derived non-informative priors are of an unknown form, the propriety of the posterior distributions is assessed to ensure admissible results. For comparison purposes, estimates obtained via the method of maximum likelihood are also considered. Finding these estimates requires solving non-linear equations, hence the Newton-Raphson algorithm is used to obtain estimates. A simulation study based on the time-to-failure of accelerated data is conducted to compare results between maximum likelihood and Bayesian estimates. As a result of the Bayesian posterior distributions being analytically intractable, two methods to obtain Bayesian estimates are considered: Markov chain Monte Carlo methods and Lindley’s approximation technique. In the simulation study the posterior means and the root mean squared error values of the estimates under the symmetric squared error loss function and the two asymmetric loss functions: the LINEX loss function and general entropy loss function, are considered. Furthermore the coverage rates for the Bayesian Markov chain Monte Carlo and maximum likelihood estimates are found, and are compared by their average interval lengths. A case study using a dataset based on accelerated time-to-failure of an insulating fluid is considered. The fit of these data for the Weibull distribution is studied and is compared to that of other popular life distributions. A full simulation study is conducted to illustrate convergence of the proper posterior distributions. Both maximum likelihood and Bayesian estimates are found for these data. The deviance information criterion is used to compare Bayesian estimates between the prior distributions. The case study is concluded by finding reliability estimates of the data at use-stress levels