On locally optimal nonparametric tests

There is an abundance of problems in which no parametric model realistically describes the situation and in which, accordingly, there is a need to resort to nonparametric methods. As the numerical problems connected with nonparametric tests are becoming less and less important, rank tests, permutation tests and the like are becoming more and more part of the standard armoury of applied statisticians. The lack of tabulated critical values, for instance, should no longer be a serious objection against the use of permutation tests in practice. The rationale underlying permutation and rank tests has been outlined in quite a number of text books and papers. Roughly speaking, permutation tests are constructible if the data can be condensed by means of a sufficient and complete statistic allowing for the proper kind of conditioning. Rank tests arise if the underlying problem is invariant with respect to (w.r.t.) a large group of transformations which leads to a maximal invariant statistic consisting of (signed) ranks. Most practical nonparametric problems, however, are too complex to be tractable by just one of those approaches. Many of them, however, can be handled by a combination of both techniques. In this paper the authors outline the logic underlying that combined reduction method and apply it to construct locally most powerful tests. Moreover, they discuss what is labelled ‘Hoeffding’s transfer problem’, i.e. the uniformly aspect of locally most powerful tests with respect to the starting point at the boundary. The authors are concentrating on the discussion of the nonparametric two-sample location and scale problem. Further important problems are mentioned in Section III.