Asymptotic Normality and Efficiency of Certain Nonparametric Test Statistics

Let X₁, , Xₘ and Y₁, , Yₙ be ordered observations from the absolutely continuous cumulative distribution functions F (x) and G (x) respectively. If z₍₈ = 1 when the ith smallest of N = m + n observations is an X and z₍₈ = 0 otherwise, then many nonparametric test statistics are of the form mTN = N₈ = ₁ E₍₈z₍₈. Theorems of Wald and Wolfowitz, Noether, Hoeffding, Lehmann, Madow, and Dwass have given sufficient conditions for the asymptotic normality of TN. In this paper we extend some of these results to cover more situations with F G. In particular it is shown for all alternative hypotheses that the Fisher-Yates-Terry-Hoeffding c₁-statistic is asymptotically normal and the test for translation based on it is at least as efficient as the t-test.