Nonparametric Bayes estimation in repair models

Consider a sequence of dependent random variables _X1,_X2,…,_Xn$X_1,X_2,\dots ,X_n$, where _X1$X_1$ has distribution F (or probability measure P  ), and the distribution of _Xi+1$X_{i+1}$ given _X1,…,_Xi$X_1,\dots ,X_i$ and other covariates and environmental factors depends on F   and the previous data, i=1,…,n-1$i=1,\dots ,n-1$. 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”.