Computing tolerance limits for the lifetime of a k-out-of-n:F system based on prior information and censored data

Exact guaranteed-coverage and expected-coverage Bayesian tolerance limits for the lifetime distribution of a k-out-of-n:F system are computed by solving nonlinear equations. The bounds are based on exponential component test data and available prior information concerning the expected component lifetime which is described by an inverted gamma distribution. The Bayesian tolerance limits are valid for single (right or left), double and progressive (standard or general) censoring, and even have frequentist validity in the noninformative case. The derived results allow the reliability engineer to judge the quality of a system prior to assembly, which offers obvious practical and economic benefits. Minimum and expected percentages of conforming systems are assessed by constructing suitable tolerance limits. Even though the viewpoints are different, the Bayesian tolerance limits that adopt the natural diffuse prior coincide numerically with recently published conditional tolerance limits in the double censoring case. The proposed Bayesian approach may be deemed as an extension of the existing frequentist methodology under double censoring that also takes into account the presence of prior information and general progressive censoring. The perspective developed simplifies and unifies the computation of tolerance limits with both frequentist and Bayesian interpretations, and also provides a probabilistic way of updating the tolerance limits in the light of new, relevant data, which is especially important in the dynamic analysis of a sequence of data. Moreover, the Bayesian approach is shown to outperform the frequentist viewpoint in terms of accuracy. In most situations, the use of substantial prior information significantly increases the accuracy level and considerably reduces the required number of failures to attain a specified degree of accuracy. Two illustrative numerical examples are studied, including the analysis of a system of water pumps for cooling a reactor. The results developed are extended to the Weibull case with unknown scale parameter and other probability models.