Likelihood Ratio Tests for Sequential $k$-Decision Problems

Abstract

Sequential tests of separated hypotheses concerning the parameter $\theta$ of a Koopman-Darmois family are studied from the point of view of minimizing expected sample sizes pointwise in $\theta$ subject to error probability bounds. Sequential versions of the (generalized) likelihood ratio test are shown to exceed the minimum expected sample sizes by at most $M \log \log \underline{\alpha}^{-1}$ uniformly in $\theta$, where $\underline{\alpha}$ is the smallest error probability bound. The proof considers the likelihood ratio tests as ensembles of sequential probability ratio tests and compares them with alternative procedures by constructing alternative ensembles, applying a simple inequality of Wald and a new inequality of similar type. A heuristic approximation is given for the error probabilities of likelihood ratio tests, which provides an upper bound in the case of a normal mean.