Hypothesis testing problems in which the null hypothesis is simple, the parameter space is finite dimensional and the supports of the probability measures are independent of the parameter are considered. Essentially complete class results are obtained for characterizing the limits of Bayes tests. Conditions for tests to be admissible and the class to be complete are given. Results are then specialized to exponential families, along with some illustrative examples.
"Complete Class Results for Hypothesis Testing Problems with Simple Null Hypotheses." Ann. Statist. 17 (1) 209 - 235, March, 1989. https://doi.org/10.1214/aos/1176347012