Asymptotically optimal multistage tests for iid data

The problem of testing two simple hypotheses about the distribution of iid random elements is considered. In particular, the focus is on multistage tests that control the two error probabilities below arbitrary, user-specified levels. A novel multistage test is proposed, analyzed, and shown to achieve the optimal expected sample size under both hypotheses, in the class of all sequential tests with the same error control, to a first-order approximation as the two target error probabilities go to zero at arbitrary rates. The proposed test is compared, both theoretically and numerically, with a multistage test that enjoys the same asymptotic optimality property under one of the two hypotheses, while performing much worse under the other.