Some Simple Examples and Counterexamples about the Existence of Optimum Tests
Abstract Theory of testing hypotheses (see [1]) derives sufficient conditions for the existence of optimum tests. These conditions often require that the class of distributions in question be ordered in some way, e.g., stochastically ordered, monotone likelihood ratio, Polya of some order, or exponential class. This article gives four simple examples which show that the sufficient ordering restrictions cannot be weakened to less stringent ordering restrictions and which thus give some insight into the structure and use of such orderings. A fifth example shows that ordering properties are not necessary for the existence of optimal tests.