Significance Probability Bounds for Rank Orderings

Abstract

The problem of two-sample rank order tests is examined from the point of view of I. R. Savage (1956, 1957, 1959, 1962). Under suitable restriction of the class of alternatives some rank orderings are always more probable than (dominated by) others. Hence, if the rejection region of a test contains the ordering $c$ it must also contain all orderings dominated by $c$ if the test is to be admissible. Thus, by counting the number of orderings dominated by $c$ we arrive at a lower bound for the size of an admissible test which rejects when $c$ is observed. Similar reasoning leads to an upper bound. The counting is achieved by putting all orderings in one-one correspondence with paths on a grid; all paths lying below or along the observed path correspond to the orderings dominated by the observed ordering. An expression for this number of paths is obtained. This expression is used to compute significance bounds for a pair of illustrative examples. Finally, the main result for the two-sample problem is extended to obtain upper and lower bounds for a one-sample problem.