Order restricted classical and Bayesian inference of a multiple step-stress model from two-parameter Rayleigh distribution under Type I censoring

This article considers a multiple step-stress model under two-parameter Rayleigh distribution with Type I censoring. Given that the lifetime of the experiment unit gradually decreases as the stress level grows step by step, the parameters of lifetime distribution with the increasing level have a natural order restriction. The reparametrization is applied to deal with this order limitation. Based on the proportional hazard model, the continuous cumulative distribution function and the corresponding likelihood function are inferred. The procedures of computing order restricted maximum likelihood estimations whether at least one failure exists in each stress level or not are introduced. Through transformation, the asymptotic confidence intervals of the original parameters are calculated based on the observed Fisher information matrix. Furthermore, taking the square error loss function, the Linex loss function, and the general entropy loss function into account, Bayesian estimations are discussed. With the importance sampling, the unknown parameters' estimates are obtained and the associated highest posterior density credible intervals are built up. With simulations under different circumstances, the effectiveness of each method proposed is demonstrated. Finally, an analysis of a real dataset is provided.