One-Sample and Paired-Sample Inferences Based on Signed Ranks

Abstract

In Chap. 2 we saw that the sign test is the best possible test (in strong senses of “best”) at level ά for a null hypothesis which is as inclusive as the statement “the observations are a random sample from a population with median 0 (or ξ₀).” It certainly seems as though better use could be made of the observations by taking their magnitudes into account. However, since the sign test is optimum at level a for this inclusive set of null distributions, any procedure which considers magnitudes would be a better test at level a only for some smaller and more restrictive set of null distributions. Such procedures are of special relevance if the restricted set is the one of interest anyway, or if the “restriction” of the null hypothesis can reasonably be assumed as a part of the model, so that the restricted hypothesis essentially amounts to the unrestricted one above. Furthermore, their exact levels may vary only slightly under the kinds of departure from assumptions which are likely, and their increased power may well be worth the small price. Recall that, in principle, level and power considerations should always be balanced off.