Bayesian Order-Restricted Inference of a Weibull Multi-Step Step-Stress Model

Abstract

Standard life-testing experiments are not appropriate for testing highly reliable units as they often turn out to be time-consuming and expensive under normal operating conditions. Under such scenario, accelerated life test is often employed to obtain failure time data. In this paper, we consider a multi-step step-stress accelerated life testing (SSALT) model when the data are type-II censored. The lifetime distribution of the experimental units at each stress level is assumed to follow a two-parameter Weibull distribution. Further, the distributions under each of the stress levels are connected through a failure rate-based multi-step SSALT model. In a step-stress experiment, the mean lifetime of the experimental units is expected to shorten with elevation in stress levels. Taking this into account, the main aim of this paper is to develop the order-restricted inference of the model parameters in a multi-step step-stress setup in a Bayesian approach. Although for the given shape parameter, the order-restricted maximum likelihood estimators (MLEs) of the model parameters can be obtained explicitly, they are in complicated forms. Additionally, the exact joint distribution of the MLEs and hence the exact CIs cannot be obtained. Under such scenario, Bayesian choice seems to be a natural one. Prior information about the model parameters is incorporated through a flexible multivariate prior distribution and it preserves the ordering of the mean lifetimes in the Bayes estimates (BEs). We propose to use the importance sampling algorithm to obtain the BEs and the associated credible intervals (CRIs). An extensive simulation study has been carried out, and our approach is illustrated with two real data sets. A novel optimal plan is proposed to determine the stress-changing time points based on the sum of the expected posterior variances.