Estimating a Survival Curve when New Is Better Than Used

Abstract

Let F be a distribution function on (0, ∞), and let S = 1 − F be its corresponding survival function. F is New Better than Used (NBU) if S(x)S(y) ≥ S(x + y) for all x and y. Let Sn(x) be the empirical survival function based on a random sample of size n from an NBU distribution function F. This paper studies the estimator Ŝn(x) defined as sup{Sn(x + y)/Sn(y)}, where the supremum is taken over all y for which Sn(y) > 0. We show that Ŝn is an NBU survival curve, and that it is strongly uniformly consistent for S when the underlying distribution has compact support (for example, when sampling is subject to type I censoring). Moreover, in such problems, we show that the rate of convergence of Ŝn is optimal.