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Model Misspecification in Discrete Time Bayesian Online Change Detection

Author

Listed:
  • Savas Dayanik

    (Bilkent University)

  • Semih O Sezer

    (Sabancı University)

Abstract

We revisit the classical formulation of the discrete time Bayesian online change detection problem in which the common distribution of an observed sequence of random variables changes at an unknown point in time. The objective is to detect the change with a stopping time of the observations and minimize a given Bayes risk. When the change time has a zero-modified geometric prior distribution, the first crossing time of the odds-ratio process over a threshold is known to be an optimal solution. In the current paper, we consider a modeler who misspecifies some of the elements of this formulation. Because of this misspecification, the modeler computes a wrong stopping threshold and follows an incorrect odds-ratio process in implementation. To find her actual Bayes risk, which is different from the value function evaluated with the wrong choices, one needs to compute the expected costs accumulated by the true odds-ratio process until modeler’s odd-ratio process crosses this wrong boundary. In the paper, we carry out these computations in the extended state space of both processes, and we illustrate these computations on examples. In those examples, we construct tolerance regions for the parameters to be estimated by the modeler. For a given choice by the modeler, the tolerance region is the set of true values for which her relative loss is less than or equal to a predetermined level.

Suggested Citation

  • Savas Dayanik & Semih O Sezer, 2023. "Model Misspecification in Discrete Time Bayesian Online Change Detection," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-27, March.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:1:d:10.1007_s11009-023-09991-4
    DOI: 10.1007/s11009-023-09991-4
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    References listed on IDEAS

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    1. Blanchet-Scalliet, Christophette & Diop, Awa & Gibson, Rajna & Talay, Denis & Tanre, Etienne, 2007. "Technical analysis compared to mathematical models based methods under parameters mis-specification," Journal of Banking & Finance, Elsevier, vol. 31(5), pages 1351-1373, May.
    2. Aleksey S. Polunchenko & Alexander G. Tartakovsky, 2012. "State-of-the-Art in Sequential Change-Point Detection," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 649-684, September.
    3. Savas Dayanik & Christian Goulding & H. Vincent Poor, 2008. "Bayesian Sequential Change Diagnosis," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 475-496, May.
    4. Bayraktar, Erhan & Dayanik, Savas & Karatzas, Ioannis, 2005. "The standard Poisson disorder problem revisited," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1437-1450, September.
    5. Savas Dayanik & Semih Onur Sezer, 2006. "Compound Poisson Disorder Problem," Mathematics of Operations Research, INFORMS, vol. 31(4), pages 649-672, November.
    6. A. N. Shiryaev & M. V. Zhitlukhin & W. T. Ziemba, 2015. "Land and stock bubbles, crashes and exit strategies in Japan circa 1990 and in 2013," Quantitative Finance, Taylor & Francis Journals, vol. 15(9), pages 1449-1469, September.
    7. Gapeev, Pavel V., 2005. "The disorder problem for compound Poisson processes with exponential jumps," LSE Research Online Documents on Economics 3219, London School of Economics and Political Science, LSE Library.
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