This paper deals with the characterization of admissibility for the testing of hypotheses, where the null hypothesis is simple, the parameter space is finite dimensional and the supports of the probability distributions do not depend on the parameter. Essentially complete class results are given for characterizing the limits of Bayes tests. Conditions for tests to be admissible and the class to be complete are given.
The given results are specialized to the exponential families with illustrative examples: testing the correlation is 0 in a bivariate normal distribution when the means and variances are known, testing the location parameter of the shifted double exponential distribution is 0, and testing that a bivariate normal distribution has zero means versus the alternative that one mean is at least as large as the other in absolute value.