A Proposed Two-Sample Rank Test: The Psi Test and its Properties

Abstract

Summary Since the power of the Psi test proposed by Gibbons (1964) compares favourably with that of other one-sided, two-sample rank tests on the equality of two distribution functions, its properties merit further investigation. This paper is devoted to such a study. A table of exact critical values for the test statistic is given for selected small sample sizes, and compared with critical values for the approximate distribution which is a function of Student's t. The agreement is good even for small samples. Using the results of Chernoff and Savage (1958), the asymptotic distribution is found to be normal in both the null and alternative cases. The null mean and variance are 0 and rπ 2l(3mN) respectively. Expressions are given for the alternative moments. The relation of the Psi test to other test statistics, specifically Terry's c 1 test, the Wilcoxon or Mann-Whitney test, and Savage's DN statistic, is investigated in terms of correlation coefficients in the null case. The test is most similar to Terry's c 1 statistic.