The Asymptotic Normality of Two Test Statistics Associated with the Two-Sample Problem

Abstract

In this paper we prove the asymptotic normality of two statistics which have been proposed to test the hypothesis that two samples come from the same parent population. One statistic is the number of runs of $X$'s and $Y$'s in the combined sample of $X$'s and $Y$'s; the other is the sum of squares of "$S_i$'s" where $S_i$ is the number of $X$'s falling between the $i$th and $(i - 1)$st largest $Y$'s. Both statistics have been studied previously, both lead to consistent tests, and both were known to be asymptotically normal under the null distribution. Here we prove limiting normality under a fairly wide class of alternatives. By means of limiting power against a sequence of alternatives which approach the null hypothesis, we compare these tests with one another and with the Smirnov test based on the sample c.d.f.'s. Against a rather large class of alternatives, the Smirnov test is seen to be considerably more powerful. The method of proving limiting normality used here is based on studying conditional moments and can be used to prove limiting normality of "combinatorial" statistics other than the ones studied herein.