Nonparametric Bayesian Estimation of Reliabilities in a Class of Coherent Systems

Abstract

Usually, methods evaluating system reliability require engineers to quantify the reliability of each of the system components. For series and parallel systems, there are limited options to handle the estimation of each component's reliability. This study examines the reliability estimation of complex problems of two classes of coherent systems: series-parallel, and parallel-series. In both of the cases, the component reliabilities may be unknown. We developed estimators for reliability functions at all levels of the system (component and system reliabilities). The main assumption required is that, for all the distributions of the components of a particular system, the sets of discontinuity points have to be disjoint. Nonparametric Bayesian estimators of all sub-distribution and distribution functions are derived, and a Dirichlet multivariate process as a prior distribution is considered for the nonparametric Bayesian estimation of all distributions. For illustration, two simulated numerical examples are presented. The estimators are s-consistent, and one may observe from the examples that they have good performance. Our estimator can accommodate continuous failure distributions, as well as distributions with mass points.