Asymptotically Optimal Sequential Testing of Multiple Hypotheses for Nonhomogeneous Gaussian Processes in Asymmetric Case

Verdenskaya, N. V.; Tartakovskii, Alexander G

doi:10.1137/1136061

Cited by 12 publications

(11 citation statements)

References 5 publications

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“…Lorden (1977) derived multiple hypothesis tests that were nearly optimal and provided a framework for further developments. Golubev and Khas'minskii (1983) discussed tests for signals embedded in Gaussian noise, and tests for other particular signal types were considered in Verdenskaya and Tartakovskii (1991), Veeravalli and Baum (1995), Tartakovsky (1998aTartakovsky ( , 1998b. Baum and Veeravalli (1994) discussed the relationship between Bayesian optimality and the multihypothesis sequential ratio test (MSPRT) that we will consider in this paper, and they showed why tests that achieve optimality for all ERs are difficult to design, if they exist at all.…”

Section: Introduction and Reviewmentioning

confidence: 99%

The dynamics of choice among multiple alternatives

McMillen

Holmes

2006

Journal of Mathematical Psychology

138

175

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Section: Introduction and Reviewmentioning

confidence: 99%

The dynamics of choice among multiple alternatives

McMillen

Holmes

2006

Journal of Mathematical Psychology

138

175

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“…This test represents a modification of the matrix SPRT (the combination of one-sided SPRT's) that was suggested by Lorden [25]. It was considered earlier by Armitage [1], Lorden [25], Dragalin [9]- [11], Verdenskaya and Tartakovskii [38], Tartakovsky [32]- [34]. For the zero-one loss function, the stopping times may be rewritten as where i.e., is the generalized weighted likelihood ratio between and the remaining hypotheses.…”

Section: And (22)mentioning

confidence: 99%

“…The first test, which is motivated by a Bayesian framework, was considered by Baum and Veeravalli [4], [36], Golubev and Khas'minskii [18], Fishman [16], Tartakovsky [31], [32] in various contexts. The second test which is formed by a specific combination of one-sided SPRT's was suggested by Armitage [1], and studied further in [9]- [11], [25], [29]- [38]. A comparison of the pros and cons of the two tests was given by us in [13] and is also summarized in Section II.…”

Section: Introductionmentioning

confidence: 99%

Multihypothesis sequential probability ratio tests. II. Accurate asymptotic expansions for the expected sample size

Dragalin¹,

Tartakovsky

Veeravalli

2000

IEEE Trans. Inform. Theory

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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.

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“…Moreover, existing research indicates that even if such a test exists, it would be very difficult to find its structure. For this reason a substantial part of the development of sequential multihypothesis testing has been directed toward the study of practical, suboptimal sequential tests and the evaluation of their asymptotic performance, see [3], [5]- [11], [21], [25], [30]- [38], and many others. A model for studying such asymptotics was first introduced by Chernoff [5], wherein he studied the independent and identically distributed (i.i.d.)…”

mentioning

confidence: 99%

“…Generalization to non-i.i.d. cases (where loglikelihood ratios fail to be random walks) have been made by Golubev and Khas'minskii [18], Lai [23], Tartakovsky [31], [32], [34], [35], as well as Verdenskaya and Tartakovskii [38].…”

mentioning

confidence: 99%

Multihypothesis sequential probability ratio tests .I. Asymptotic optimality

Draglia

Tartakovsky

Veeravalli

1999

IEEE Trans. Inform. Theory

259

241

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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.

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Asymptotically Optimal Sequential Testing of Multiple Hypotheses for Nonhomogeneous Gaussian Processes in Asymmetric Case

Cited by 12 publications

References 5 publications

The dynamics of choice among multiple alternatives

The dynamics of choice among multiple alternatives

Multihypothesis sequential probability ratio tests. II. Accurate asymptotic expansions for the expected sample size

Multihypothesis sequential probability ratio tests .I. Asymptotic optimality

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