Testing Whether New is Better than Used

A U-statistic Jₙ is proposed for testing the hypothesis H₀ 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 H₁ that a new item has stochastically greater life length (F̄(x)F̄(y) F̄(x + y), for all x 0, y 0, where F̄ = 1 - F). Jₙ is unbiased; in fact, under a partial ordering of H₁ distributions, Jₙ is ordered stochastically in the same way. Consistency against H₁ 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.