Estimating system reliability under Brown–Proschan imperfect repair with covariates

We propose an imperfect repair model which depends on external effects quantified by covariates. The model is based on the Brown–Proschan (BP) imperfect repair model wherefrom the probability of perfect repair is represented by a function of covariates. We are motivated by deficiency of the BP model whose stationarity prevents us from predicting dynamically the time to next failure according to external condition. Five types of function for the probability of perfect repair are proposed. This article also presents a procedure for estimating the parameter of the function for the probability of perfect repair, as well as the inherent lifetime distribution of the system, based on consecutive inter-failure times and the covariates. The estimation procedure is based on the expectation-maximization principle which is suitable to incomplete data problems. Focusing on the maximization step, we derive some theorems which guarantee the existence of the solution. A Monte Carlo study is also performed to illustrate the prediction power of the model as well as to show reasonable properties of the estimates. The model reduces significantly the mean square error of the in-sample prediction, so it can be utilized in real fields for evaluating and maintaining repairable systems.