Nonparametric Bayes estimation of the reliability function of a coherent system

Complex coherent systems are the engines driving forward our technological world. A coherent system is composed of components, which could be modules or sub-systems, that interact with each other according to some structure function. For purposes of maintenance and safety considerations, it is of critical importance to gain knowledge of the distribution of the system lifetime, with this distribution being a function of the distributions of the components lifetimes. Since the monitoring of a system ceases upon system failure, at system failure some components will be failed, while others, depending on the structure function, will still be functioning with their lifetimes right-censored by the system lifetime. This paper deals with the estimation of the system lifetime distribution. The inferential framework is nonparametric Bayesian, with partition-based Dirichlet processes (PBDP) assigned as priors on the components lifetime distributions. PBDP are more general than the usual Dirichlet process (DP) priors and are particularly suited as priors in settings with censored data. The resulting estimator of the system life distribution, which is a function of the nonparametric Bayes estimators of the components lifetime distributions, is compared in terms of bias and variance with a product-limit type estimator proposed by Doss, et. al. (Ann. Statist., 1989), which can be obtained as a limit of the proposed estimator. These comparisons, which are facilitated through computer simulations, demonstrate that the proposed estimator possesses some robustness. The proposed estimator is illustrated using a synthetic data for a parallel system with five components.