Global and Partial Non-Nested Hypotheses and Asymptotic Local Power

This paper addresses two related issues in the literature of non-nested hypotheses testing. Firstly, by means of a measure of “closeness” of probability density functions, it shows how <italic>any</italic> two hypotheses can be placed into the nested and the non-nested categories with the latter category being subdivided further into “globally” and “partially” non-nested hypotheses. Secondly, by emphasizing the distinction between a “local null” and a “local alternative,” the paper shows that only in the case of partially non-nested hypotheses is it possible to specify local alternatives. In this case the paper derives the asymptotic distribution of the Cox test statistic under local alternatives and shows that it is distributed as a normal variate with a mean which is directly related to the measure of “closeness” of the alternative to the null hypothesis.