On Weak Convergence of an Estimator of the Survival Function When New is Better Than Used of a Specified Age

原文摘要

A survival function S is said to be in the New Better than Used of age t 0 (NBU-t 0) class if S(x + t 0) ≤ S(t 0)S(x) for all x ≥ 0. Reneau and Samaniego proposed an estimator Ŝ n for S when S is known as a member of the NBU-t 0 class. Many properties of Ŝ n were studied by Reneau and Samaniego. The problems of the weak convergence of Wn = √n (Ŝ n − S), however, was not solved. In this article, we show that the weak convergence of Wn does not hold in general and establish sufficient conditions for the weak convergence to hold. Three important cases are as follows: (1) If the underlying survival function S is from the subclass of New Strictly Better than Used of age t 0 (e.g., gamma and Weibull distributions), then Wn converges weakly with the same limiting distribution as that of the empirical process. In this case, confidence bands for S can be easily constructed. (2) If S is from the subclass of New is the Same as Used of age t 0 (e.g., exponential distributions), then the weak convergence of Wn holds, and the finite-dimensional limiting distribution is estimable. Wn does not converge weakly to a Brownian bridge, however. (3) If S satisfies neither (1) nor (2), then the weak convergence of Wn fails to hold. Examples for this case are given in the last section.

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