Wherefore similar tests?

Similarity of a test is often a necessary condition for a test to be unbiased (in particular for a test to be uniformly most powerful unbiased when such a test exists). Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1986) describes the connection between similar tests and uniformly most powerful unbiased tests. The methods to achieve these properties as outlined in Lehmann are used extensively. In any case, an admissible similar test is frequently one that can be recommended for practical use. In some constrained parameter spaces however, we show that admissible similar tests sometimes completely ignore the constraints. In some of these cases we call such tests constraint insensitive. The tests seem not to be intuitive and perhaps should not be used. On the other hand, there are models with constrained parameter spaces where similar tests do take into account the constraints. In these cases the admissible test is called constraint sensitive. We offer a systematic approach that enables one to determine whether an admissible similar test is constraint insensitive or not. The approach is applied to three classes of models involving order restricted parameters. The models include testing for homogeneity of parameters, testing subsets of parameters, and testing goodness of fit of a family of discrete distributions.