Hierarchical models in repairable system reliability analysis

Suppose we observe a number of repairable systems, and each system has multiple failures throughout its life. A reasonable model for this depends on how similar we believe these systems are. One possibility is to assume that the systems share no similarities and are completely unalike. In this situation, we would analyze each system separately. Even if we assume the same kind of model, e.g., a nonhomogeneous Poisson process, we would use only data from system i to estimate the parameters for system i. At the other end of the spectrum, we might assume that all systems are identical; the model, and even the parameters of the model, are the same for all systems. In the middle, we might assume that the systems are similar, but not identical, in the sense that the parameters of the models for the multiple systems can be thought of as coming from some prior distribution. We cannot observe these parameters directly, but the observed failure times for system i depend on the parameters for system i, so we obtain indirect information about each parameter. We discuss three paradigms for analyzing such hierarchical models: naive empirical Bayes, hierarchical Bayes, and likelihood methods using data cloning.