A unified theory of hypothesis testing based on rankings.

Abstract

A unified theory of hypothesis testing based on the ranks of the data is proposed. A hypothesis testing problem often gives rise to two separate permutation sets corresponding to the data and to the alternative, respectively. By defining the distance between permutation sets as the average of all distances between pairs of permutations, one from each set, it is possible to obtain various test statistics. The limiting distributions of test statistics derived by the unified approach herein are obtained under both the null hypothesis and contiguous alternatives. The unified approach produces not only some well-known test statistics but also some new yet plausible test statistics. The corresponding results are extensions of the simple linear rank statistics defined by Hajek and Sidak (1967) to the generalized linear rank statistics and of the two-sample case to the multi-sample case. Furthermore, a combined method was developed for the case of composite alternatives.