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Multisource Bayesian sequential binary hypothesis testing problem

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  • Savas Dayanik
  • Semih Sezer

Abstract

We consider the problem of testing two simple hypotheses about unknown local characteristics of several independent Brownian motions and compound Poisson processes. All of the processes may be observed simultaneously as long as desired before a final choice between hypotheses is made. The objective is to find a decision rule that identifies the correct hypothesis and strikes the optimal balance between the expected costs of sampling and choosing the wrong hypothesis. Previous work on Bayesian sequential hypothesis testing in continuous time provides a solution when the characteristics of these processes are tested separately. However, the decision of an observer can improve greatly if multiple information sources are available both in the form of continuously changing signals (Brownian motions) and marked count data (compound Poisson processes). In this paper, we combine and extend those previous efforts by considering the problem in its multisource setting. We identify a Bayes optimal rule by solving an optimal stopping problem for the likelihood-ratio process. Here, the likelihood-ratio process is a jump-diffusion, and the solution of the optimal stopping problem admits a two-sided stopping region. Therefore, instead of using the variational arguments (and smooth-fit principles) directly, we solve the problem by patching the solutions of a sequence of optimal stopping problems for the pure diffusion part of the likelihood-ratio process. We also provide a numerical algorithm and illustrate it on several examples. Copyright Springer Science+Business Media, LLC 2012

Suggested Citation

  • Savas Dayanik & Semih Sezer, 2012. "Multisource Bayesian sequential binary hypothesis testing problem," Annals of Operations Research, Springer, vol. 201(1), pages 99-130, December.
    • Handle: RePEc:spr:annopr:v:201:y:2012:i:1:p:99-130:10.1007/s10479-012-1217-z
    DOI: 10.1007/s10479-012-1217-z
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    References listed on IDEAS

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    1. Dayanik, Savas & Sezer, Semih O., 2006. "Sequential testing of simple hypotheses about compound Poisson processes," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1892-1919, December.
    2. René Carmona & Savas Dayanik, 2008. "Optimal Multiple Stopping of Linear Diffusions," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 446-460, May.
    3. Dayanik, Savas & Karatzas, Ioannis, 2003. "On the optimal stopping problem for one-dimensional diffusions," Stochastic Processes and their Applications, Elsevier, vol. 107(2), pages 173-212, October.
    4. Savas Dayanik, 2008. "Optimal Stopping of Linear Diffusions with Random Discounting," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 645-661, August.
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    Cited by:

    1. Pavel V. Gapeev, 2020. "On the problems of sequential statistical inference for Wiener processes with delayed observations," Statistical Papers, Springer, vol. 61(4), pages 1529-1544, August.
    2. Belleh Fontem, 2022. "An optimal stopping policy for car rental businesses with purchasing customers," Annals of Operations Research, Springer, vol. 317(1), pages 47-76, October.

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