A simple and effective scanning rule for a multi-channel system

Scanning rule, allocation rule, SPRT, change point problem, CUSUM procedure,

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 according to one of the underlying distributions, and the decision risks go to zero. Simulation results for practical examples show that these approximations ar..

Multihypothesis sequential probability ratio tests -- Part I: Asymptotic optimality

The problem of sequential testing of multiple hypotheses is considered, and two candidate sequential test procedures are studied. Both tests are multihypothesis versions of the binary sequential probability ratio test (SPRT), and are referred to as MSPRT’s. The first test is motivated by Bayesian optimality arguments, while the second corresponds to a generalized likelihood ratio test. It is shown that both MSPRT’s are asymptotically optimal relative not only to the expected sample size but also to any positive moment of the stopping time distribution, when the error probabilities or, more generally, risks associated with incorrect decisions are small. The results are first derived for the discrete-time case of independent and identically distributed (i.i.d.) observations and simple hypotheses. They are then extended to general, possibly continuous-time, statistical models that may include correlated and nonhomogeneous observation processes. It also demonstrated that the results can be extended to hypothesis testing problems with nuisance parameters, where the composite hypotheses, due to nuisance parameters, can be reduced to simple ones by using the principle of invariance. These results provide a complete generalization of the results given in [36], where it was shown that the quasi-Bayesian MSPRT is asymptotically efficient with respect to the expected sample size for i.i.d. observations. In a companion paper [12], based on the nonlinear renewal theory we find higher order approximations, up to a vanishing term, for the expected sample size that take into account the overshoot over the boundaries of decision statistics