Chapter 3 - Elementary theory of rank tests

Abstract

This chapter examines the elementary theory of rank tests. Rank tests may be obtained by the process of specialization from either of the two families of tests. The tests include permutation tests and the tests invariant under changes of location and scale. The ranks are well defined only if the probability of coincidence of any pair of coordinates equals zero. It is found that for similar tests the probability of the error of the first kind is constant throughout the whole hypothesis. The most powerful permutation tests may be found quite easily, at least theoretically, because, by the conditional argument, the problem may be carried over from the λ-field of similar events to the σ-field of all subsets. It is observed that if the alternative is composite, then the family of rank tests does not contain the asymptotically maximin most powerful test in many important cases where the family of permutation tests does. It is found that rank tests are invariant not only under increasing linear transformations but under all strictly increasing continuous transformations and thus, constitute a subfamily of tests invariant under location and scale changes.