General Theory of Hypothesis Testing

Abstract

The notion of distance was fruitfully utilized in previous chapters in order to develop tests of hypotheses for both complete and incomplete rankings. In this chapter we consider a more general framework for constructing tests of hypotheses. We begin by defining two sets of rankings: one set consists of all the rankings which are most in agreement with the observed ranking while the second set contains all the rankings which are most in agreement with the alternative hypothesis. A distance function is then defined between those two sets of rankings. The notion of distance between sets is well known in mathematics and is often taken to be the minimum distance between pairs of elements, one from each set. In the present statistical context however, the more workable definition of distance is chosen to be the average of all pairwise distances between pairs of rankings, one from each set.

Chapter Notes

Alvo and Pan (1997) also discussed the situation when the alternatives are unordered by considering the union of the r! possible ordered alternatives.

For the problem of testing for umbrella alternatives, we refer the reader to Alvo (2008) for additional references and for a brief history of the subject. The Spearman statistic considers the data on either side of the peak separately whereas the Kendall statistic (6.15) considers, in addition, the relationship of the data between both sides of the peak. For small sample sizes this may increase the sensitivity of that statistic. The approach presented may have potential applications in the study of isotonic regression.