Estimating system reliability with fully masked data under Brown-Proschan imperfect repair model

Abstract This article presents a statistical procedure for estimating the lifetime distribution of a repairable system based on consecutive inter-failure times of the system. The system under consideration is subject to the Brown-Proschan imperfect repair model. The model postulates that at failure the system is repaired to a condition as good as new with probability p , and is otherwise repaired to its condition just prior to failure. The estimation procedure is developed in a parametric framework for incomplete set of data where the repair modes are not recorded. The expectation-maximization principle is employed to handle the incomplete data problem. Under the assumption that the lifetime distribution belongs to the two-parameter Weibull family, we develop a specific algorithm for finding the maximum likelihood estimates of the reliability parameters, the probability of perfect repair ( p ), as well as the Weibull shape and scale parameters (α, β) The proposed algorithm is applicable to other parametric lifetime distributions with aging property and explicit form of the survival function, by just modifying the maximization step. We derive some lemmas which are essential to the estimation procedure. The lemmas characterize the dependency among consecutive lifetimes. A Monte Carlo study is also performed to show the consistency and good properties of the estimates. Since useful research is available regarding optimal maintenance policies based on the Brown-Proschan model, the estimation results will provide realistic solutions for maintaining real systems.