Nearly optimal sequential tests of composite hypotheses revisited

We revisit the problem of sequential testing composite hypotheses, considering multiple hypotheses and very general non-i.i.d. stochastic models. Two sequential tests are studied: the multihypothesis generalized sequential likelihood ratio test and the multihypothesis adaptive sequential likelihood ratio test with one-stage delayed estimators. While the latter loses information compared to the former, it has an advantage in designing thresholds to guarantee given upper bounds for probabilities of errors, which is practically impossible for the generalized likelihood ratio type tests. It is shown that both tests have asymptotic optimality properties minimizing the expected sample size or even more generally higher moments of the stopping time as probabilities of errors vanish. Two examples that illustrate the general theory are presented.