Testing Whether New is Better than Used with Randomly Censored Data

A life distribution F, with survival function$\bar{F} \equiv 1 - F$, is new better than used (NBU) if F̄(x + y) ≤ F̄(x)F̄(y) for all x, y ≥ 0. We propose a test of H0: F is exponential, versus H1: F is NBU, but not exponential, based on a randomly censored sample of size n from F. Our test statistic is Jc n= ∫ ∫ F̄n(x + y) dFn(x) dFn(y), where Fnis the Kaplan-Meier estimator. Under mild regularity on the amount of censoring, the asymptotic normality of Jc n, suitably normalized, is established. Then using a consistent estimator of the null standard deviation of n1/2Jc n, an asymptotically exact test is obtained. We also study, using tests for the censored and uncensored models, the efficiency loss due to the presence of censoring.