Explicit characterizations are given of the minimal complete class and a minimal essentially complete class of tests of a simple hypothesis specifying the mean of a uniform distribution of known range. Examples are given of tests which are optimal against various alternatives.
Allan Birnbaum. "Admissible Tests for the Mean of a Rectangular Distribution." Ann. Math. Statist. 25 (1) 157 - 161, March, 1954. https://doi.org/10.1214/aoms/1177728856