Logarithmically efficient simulation for misclassification probabilities in sequential multiple testing

We consider the problem of estimating via Monte Carlo simulation the misclassification probabilities of two sequential multiple testing procedures. The first one stops when all local test statistics exceed simultaneously either a positive or a negative threshold. The second assumes knowledge of the true number of signals, say m, and stops when the gap between the top m test statistics and the remaining ones exceeds a threshold. For each multiple testing procedure, we propose an importance sampling algorithm for the estimation of its misclassification probability. These algorithms are shown to be logarithmically efficient when the data for the various statistical hypotheses are independent, and each testing problem satisfies an asymptotic stability condition and a symmetry condition. Our theoretical results are illustrated by a simulation study in the special case of testing the drifts of Gaussian random walks.