Second order asymptotic and non-asymptotic optimality properties of combined tests

Let p independent test statistics be available to test a null hypothesis concerned with the same parameter. The p are assumed to be similar tests. Asymptotic and non-asymptotic optimality properties of combined tests are studied. The asymptotic study centers around two notions. The first is Bahadur efficiency. The second is based on a notion of second order comparisons. The non-asymptotic study is concerned with admissibility questions. Most of the popular combining methods are considered along with a method not studied in the past. Among the results are the following: Assume each of the p statistics has the same Bahadur slope. Then the combined test based on the sum of normal transforms, is asymptotically best among all tests studied, by virtue of second order considerations. Most of the popular combined tests are inadmissible for testing the noncentrality parameter of chi-square, t, and F distributions. For chi-square a combined test is offered which is admissible, asymptotically optimal (first order), asymptotically optimal (second order) among all tests studied, and for which critical values are obtainable in special cases. Extensions of the basic model are given.