In an earlier paper the author proved the existence of a best test for testing hypotheses about a random parameter with unknown distribution. This paper gives a result which helps one find the best test approximately for several of the examples considered in the previous paper.
Glen Meeden. "Finding Best Tests Approximately for Testing Hypotheses About a Random Parameter." Ann. Math. Statist. 42 (4) 1452 - 1454, August, 1971. https://doi.org/10.1214/aoms/1177693261