Asymptotically optimal multistage tests for multihypothesis testing

A multistage test is proposed for the problem of testing an arbitrary number of simple hypotheses regarding the distribution of a sequence of i.i
$\mathbf{d}$
. random elements. The proposed test is shown to control the probability of each possible error under an arbitrary, user-specified level. Most importantly, it is shown to achieve the optimal expected sample size under every hypothesis, in the class of all sequential tests with the same levels of error control, to a first-order asymptotic approximation as these levels go to zero. These theoretical results are illustrated in a simulation study, where the proposed multistage test is compared with an asymptotically optimal fully-sequential test.