Sequential Multiple Testing of Multiple Composite Hypotheses: an Asymptotic Optimality Theory with General Information Functions

This paper considers the problem of testing the marginal distributions of multiple, independent data streams, where for each data stream, multiple composite hypotheses along with an indifference zone are posed.A novel global error metric is proposed, which aims to control the probabilities of making different numbers of misclassifications below different, user-specified levels, and which includes the classical and the generalized misclassification probabilities as special cases.A novel testing procedure is designed and is shown to achieve the minimum expected sample size under all possible distributions, among all tests that control this global error metric below the same levels, asymptotically as any of these levels goes to zero.This asymptotic optimality theory is established allowing temporal dependence and general information functions beyond linear that are considered in most literature.Examples are provided to illustrate the theory and numerical studies are presented to visualize both the asymptotic properties and finite-sample performance.