Sufficient condition for admissibility of the Wilcoxon test in the classical two-sample problem

We consider nonparametric two-sample problems of testing the equality of two continuous distribution functions F and G. Whether or not the Wilcoxon test is admissible within the class of all tests, and whether or not the two-sided Wilcoxon test is admissible within the class of all rank tests are two longstanding open questions (Lehmann, 1959; 1986: Ferguson, 1967). In this article, we establish a sufficient condition that the Wilcoxon test is admissible in these two-sample problems. As an application, we show that for some special cases, the Wilcoxon test is admissible within the class of all tests and the two-sided Wilcoxon test is admissible within the class of all rank tests. The author believes that the sufficient condition can be used to solve the longstanding open questions in Lehmann (1959) on a case-by-case basis, but is unable to produce a unified proof for all cases.