Estimating a Survival Curve when New Is Better Than Used
Russell A. Boyles, Francisco J. Samaniego
Operations Research · 1984 · Volume 32 · Issue 3
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 S n ( 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 Ŝ n ( x ) defined as sup{ S n ( x + y )/ S n ( y )}, where the supremum is taken over all y for which S n ( y ) > 0. We show that Ŝ n 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 Ŝ n is optimal.
Keywords
723 New Better than Used (NBU) class, 726 life testing under type I censoring, 799 estimating a survival curve
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