Bayesian Accelerated Life Testing of Series Systems

Abstract

Consider life testing of J-component series systems that are subjected to stress levels that are steeper than that at normal usage condition. The objective of performing such life tests, commonly known as Accelerated Life Testing (ALT) in the literature, is to collect observations on system failure times within a limited time frame. The accelerated observations are then used to infer on the component and system reliability metrics at usage stress. In this thesis, the existing literature is first extended by considering the general case of K stress variables, as opposed to the usual consideration of a single stress variable. Next, a general model assuming that the component log-lifetimes belong to an arbitrary location-scale family of distributions, is formulated. The location parameters are assumed to depend on the stress variables through a general stress translation function, while the scale parameters are assumed to be independent of the stress variables. This formulation covers the standard lifetime distributions as well as well-known stress translation functions as special cases. Bayesian methodologies are then developed for four special cases of the proposed general model, viz., exponentials, Weibulls with equal shape parameter, Weibulls with distinct shape parameters and log-normals with distinct scale parameters. For exponential and Weibull models, the priors on lifetime parameters are assumed to be log-concave and independent of each other. The resulting univariate conditional posterior of each lifetime parameter given the rest, is shown to be log-concave. This facilitates Gibbs sampling from the joint posterior of lifetime parameters. Propriety of the joint posteriors with Laplacian uniform priors on stress coefficients are also proved under a suitable set of sufficient conditions. For the log-normal model, the observed data is first augmented with log-lifetimes of un-failed components to form complete data. A Gibbs sampling scheme is then developed to generate observations from the joint posterior of lifetime parameters, through the augmented data and a conjugate prior for the complete data. In all four cases, Bayesian predictive inference techniques are used to study component and system reliability metrics at usage stress. Though this thesis mainly deals with Bayesian inference of accelerated data of series systems, maximum likelihood analysis for the log-normal component lifetimes is also performed via an expectation-maximization (EM) algorithm and bootstrap, which are not available in the literature. The last part of this thesis deals with construction of optimal Bayesian designs for accelerated life tests of J-component series systems under Type-I censoring scheme. Optimal ALT plans for a single stress variable are obtained using two different Bayesian D-optimality criteria for exponentially distributed component lives. A detailed sensitivity analysis is carried out to investigate the effect of different planning inputs on the optimal designs as well.