Multihypothesis sequential probability ratio tests - Part II: Accurate asymptotic expansions for the expected sample size

In a companion paper [13], we proved that two specific constructions of multihypothesis sequential tests, which we refer to as Multihypothesis Sequential Probability Ratio Tests (MSPRT's), are asymptotically optimal as the decision risks (or error probabilities) go to zero, The MSPRT's asymptotically minimize not only the expected sample size but also any positive moment of the stopping time distribution, under very general statistical models for the observations. in this paper, based on nonlinear renewal theory we find accurate asymptotic approximations (up to a vanishing term) for the expected sample size that take into account the "overshoot" over the boundaries of decision statistics. The approximations are derived for the scenario where the hypotheses are simple, the observations are independent and identically distributed (i.i.d.) according to one of the underlying distributions, and the decision risks go to zero. Simulation results for practical examples show that these approximations are fairly accurate not only for large but also for moderate sample sizes. The asymptotic results given here complete the analysis initiated in [4], where first-order asymptotics were obtained for the expected sample size under a specific restriction on the Kullback-Leibler distances between the hypotheses.