Nonparametric Bayes and empirical Bayes estimators of mean residual life at age \(t\) (Q1193947)

The mean residual life (MRL) at age \(t\geq 0\) is the expected remaining life of an item, given that the item is of age \(t\): \[ M(t)=\begin{cases} \mathbb{E}(X-t|\;X>t) \quad &\text{if } \overline{F}(t)>0\\ 0 &\text{if } \overline{F}(t)=0.\end{cases} \] This paper proposes estimators of \(M(t)\), using the nonparametric Bayes and empirical Bayes methods based on \textit{T. S. Ferguson's} [Ann. Stat. 1, 209-230 (1973; Zbl 0255.62037)] Dirichlet process prior under squared error loss. In Section 2 of the paper it is stated that the Bayes estimate of \(M(t)\), under a Dirichlet process prior with a parameter \(\alpha\), has the form \[ e_ B(t)=D(t)M_ 0(t)+(1-D(t))\widehat M(t), \quad\text{where}\quad D(t)=\alpha(t,+\infty)/[\alpha(t,+\infty)+n-k], \] \(n\) is the number of observations, \(k\) is the number of observations with values less than or equal to \(t\) and \(\widehat M(t)\) is the empirical MRL estimator discussed by \textit{G. L. Yang} [Ann. Stat. 6, 112-116 (1978; Zbl 0371.62055)] and \textit{W. J. Hall} and \textit{J. A. Wellner} [Tech. Rep., Dpt. Stat. Univ. Rochester (1979)]. A modified form holds when ties are present in the observations. Section 3 is devoted to some results about empirical Bayes estimators [in particular asymptotic optimality in the sense of \textit{H. Robbins}, Proc. 3rd Berkeley Sympos. Math. Stat. Probability 1, 157-163 (1956; Zbl 0074.353)]. A couple of illuminating remarks are also given in this Section. The proofs of results stated in Sections 2 and 3 are presented in Section 4.