Bayesian nonparametric estimation in a series system or a competing-risks model

This article presents a Bayesian nonparametric approach to the estimation of a system and its components' survival functions arising from observing the failure of a series system or a competing risk model. A Dirichlet multivariate process is used as a prior for the vector of the components' random subsurvival function to derive Bayes estimator of the survival function when the cause of failure belongs to a certain risk subset. This is done as follows. First, Peterson's formula is evaluated using the Bayes estimators of the subsurvival functions corresponding to the risk subset, to obtain a plugged-in nonparametric estimator of the survival function associated with the risk subset. Then, using the product-integration approach, it is proved that this nonparametric estimator is in fact the Bayes estimator of the survival function corresponding to the risk subset under quadratic loss function and the Dirichlet multivariate process. The weak convergence and the strong consistency of the estimator is established. The special case when the system has only two components corresponds to well studied randomly censored model.