A distribution free test for the two sample problem for general alternatives

Abstract

A new distribution free test for the two sample problem is presented. The test statistic is derived from a descriptive measure of interdistributional inequality, thus having an intuitive pictorial interpretation. Some quantiles of the exact distribution of the new test statistic under H 0 are computed for balanced samples sized up to fourteen observations from each distribution. The asymptotic distribution of the test statistic under H 0 is that of the integral of the absolute value of a Brownian bridge. Using Monte Carlo simulations we found that the distribution of the test statistic is already reasonably well approximated by the asymptotic distribution for rather small sample sizes. Further, we compare the new test in terms of power against general alternatives to other well-known two sample tests, namely the Cramer-von Mises test, the Kolmogorov-Smirnov test, and the Wilcoxon-Mann-Whitney test. It turns out that the new test performs similarly as the Cramer-von Mises test. Both tests clearly dominate the Kolmogorov-Smirnov test.