Nonparametric Bayes estimation in repair models

Abstract Consider a sequence of dependent random variables X 1 , X 2 , … , X n , where X 1 has distribution F (or probability measure P ), and the distribution of X i + 1 given X 1 , … , X i and other covariates and environmental factors depends on F and the previous data, i = 1 , … , n - 1 . General repair models give rise to such random variables as the failure times of an item subject to repair. There exist nonparametric non-Bayes methods of estimating F in the literature, for instance, Whitaker and Samaniego [1989. Estimating the reliability of systems subject to imperfect repair. J. Amer. Statist. Assoc. 84, 301–309], Hollander et al. [1992. Nonparametric methods for imperfect repair models. Ann. Statist. 20, 879–896] and Dorado et al. [1997. Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160] , etc. Typically these methods apply only to special repair models and also require repair data on N independent items until exactly only one item is left awaiting a “perfect repair”. In this paper, we define a general model for dependent random variables taking values in a general space, which includes most of the repair models in the literature. We describe nonparametric Bayesian methods to estimate P , without making any assumptions on when we stop collecting data. To do this we introduce a new class of priors called partition-based (PB) priors and show that it is a conjugate class to a large class of our general repair models. We also define a subclass of such priors called partition-based Dirichlet (PBD) priors which also forms a conjugate family of priors. For a special case of the repair model called the aging repair model, we obtain an easily computable Bayes estimate of P under a Dirichlet prior. The Bayes estimates are smoother than Whitaker and Samaniego non-Bayes estimates. Graphical comparisons show that the Bayes and non-Bayes estimates tend to be close.