Goodness Criteria for Two-Sample Distribution-Free Tests

Some of the concepts and results of Chapman [3] for one-sample distribution-free tests are extended to the two-sample problem. Chapman's lead will be followed since his work gives a goodness criterion for comparing distribution-free tests, for finite sample sizes, over a large class of one-sided alternatives. Since all two-sample distribution-free statistics known to the authors satisfy Scheffe's [10] boundary condition, and all, except those of Pitman, also are strongly distribution-free (SDF) and therefore [1] are rank statistics, consideration may reasonably be restricted to rank tests. Such tests with the additional property of monotonicity are unbiased, partially ordered, and assume maximum and minimum powers for certain reasonable alternatives. Some of the maximum powers of the Mann-Whitney-Wilcoxon, Fisher-Yates, van der Waerden, Doksum, Savage, Epstein-Rosenbaum, and Cramer-von Mises statistics are tabulated. (For definitions and references, see Section 6.)