Testing Whether New is Better than Used with Randomly Censored Data

Abstract

A life distribution $F$, with survival function $\bar{F} \equiv 1 - F$, is new better than used (NBU) if $\bar{F}(x + y) \leq \bar{F}(x)\bar{F}(y)$ for all $x, y \geq 0$. We propose a test of $H_0 : F$ is exponential, versus $H_1 : F$ is NBU, but not exponential, based on a randomly censored sample of size $n$ from $F$. Our test statistic is $J^c_n = \int \int \bar{F}_n(x + y) dF_n(x) dF_n(y)$, where $F_n$ is the Kaplan-Meier estimator. Under mild regularity on the amount of censoring, the asymptotic normality of $J^c_n$, suitably normalized, is established. Then using a consistent estimator of the null standard deviation of $n^{1/2}J^c_n$, 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.