  Given a fixed-sample-size test that controls the error probabilities under
two specific, but arbitrary, distributions, a 3-stage and two 4-stage tests are
proposed and analyzed. For each of them, a novel, concrete, non-asymptotic,
non-conservative design is specified, which guarantees the same error control
as the given fixed-sample-size test. Moreover, first-order asymptotic
approximation are established on their expected sample sizes under the two
prescribed distributions as the error probabilities go to zero. As a corollary,
it is shown that the proposed multistage tests can achieve, in this asymptotic
sense, the optimal expected sample size under these two distributions in the
class of all sequential tests with the same error control. Furthermore, they
are shown to be much more robust than Wald's SPRT when applied to one-sided
testing problems and the error probabilities under control are small enough.
These general results are applied to testing problems in the iid setup and
beyond, such as testing the correlation coefficient of a first-order
autoregression, or the transition matrix of a finite-state Markov chain, and
are illustrated in various numerical studies.
