Domains of Optimality of Tests in Simple Random Sampling

Abstract

This paper deals with the structure of sets $\Omega$ of distributions for which a particular test is the most powerful for testing a simple hypothesis $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$, that is, with the domain of optimality of a test. The context is restricted to these $\Omega$ consisting of probabilities having continuous positive densities, and to one-sample tests. The important concept is that of a family of tests, one for each significance level. This concept allows us to use the full power of the Neyman-Pearson Lemma. The main results are: (1) The domain of optimality of a test family $\Phi$ is essentially a multiplicatively-convex (convex in the logarithms) cone; hence there are distributions both "near to" and "far from" the null distribution for which $\Phi$ is optimal. (Theorems 1, 2, and 3). (2) If $\Phi$ is uniformly most powerful for testing $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$ with $n \geqq 2$ then the class of distributions has a monotone likelihood ratio. (Theorem 4).