Abstract: Given a fixed-sample-size test that controls the error probabilities under two specific arbitrary distributions, we propose and analyze a 3-stage test and two 4-stage tests. For each test, we specify a novel, concrete, non-conservative design, and establish a first-order asymptotic approximation for the expected sample size under the two prescribed distributions as the error probabilities go to zero. As a corollary, we show that the proposed multistage tests can asymptotically achieve the optimal expected sample size under these two distributions in the class of all sequential tests with the same error control. Furthermore, the tests are shown to be more robust than Wald's sequential probability ratio test when applied to one-sided testing problems and the error probabilities under control are small. We apply these general results to testing problems in the independent and identically distributed setup and beyond, such as testing the correlation coefficient of a first-order autoregressive process or testing the transition matrix of a finite-state Markov chain, and illustrate them in various numerical studies.

Key words and phrases: Asymmetric error probabilities, asymptotic optimality, group-sequential tests, large-deviation theory, multistage tests, sequential testing.