Fixed sample-size procedures are considered for testing a one-sided composite hypothesis concerning a real, one-dimensional parameter of an exponential distribution (1.1). In particular, conditions are studied such that the minimax tests have a critical region which is a semi-infinite interval on the real line.
S. G. Allen Jr.. "A Class of Minimax Tests for One-sided Composite Hypotheses." Ann. Math. Statist. 24 (2) 295 - 298, June, 1953. https://doi.org/10.1214/aoms/1177729036