Testing Whether New is Better than Used

Abstract

A $U$-statistic $J_n$ is proposed for testing the hypothesis $H_0$ that a new item has stochastically the same life length as a used item of any age (i.e., the life distribution $F$ is exponential), against the alternative hypothesis $H_1$ that a new item has stochastically greater life length $(\bar{F}(x)\bar{F}(y) \geqq \bar{F}(x + y)$, for all $x \geqq 0, y \geqq 0$, where $\bar{F} = 1 - F). J_n$ is unbiased; in fact, under a partial ordering of $H_1$ distributions, $J_n$ is ordered stochastically in the same way. Consistency against $H_1$ alternatives is shown, and asymptotic relative efficiencies are computed. Small sample null tail probabilities are derived, and critical values are tabulated to permit application of the test.