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 Teor. Veroyatnost. i Primenen., 2008, Volume 53, Issue 3, Pages 610–622 (Mi tvp2456)

On the Best 2-CUSUM Stopping Rule for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model

a Columbia University
b Princeton University

Abstract: This work examines the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. Traditionally, 2-CUSUM stopping rules have been used for this problem due to their asymptotically optimal character as the mean time between false alarms tends to $\infty$. In particular, attention has focused on 2-CUSUM harmonic mean rules due to the simplicity of calculating their first moments. In this paper, expressions for the first moment of a general 2-CUSUM stopping rule and its rate of change are derived. These expressions are used to obtain explicit upper and lower bounds for it and its rate of change as one of the threshold parameters changes. Using these expressions we prove not only the existence but also the uniqueness of the best classical 2-CUSUM stopping rule with respect to the extended Lorden criterion suggested in [O. Hadjiliadis and G. V. Moustakides, Theory Probab. Appl., 50 (2006), pp. 75–85]. In particular, in both the symmetric and the nonsymmetric case we identify the class of the best 2-CUSUM stopping rule

Keywords: change detection, quickest detection, CUSUM, two-sided CUSUM.

DOI: https://doi.org/10.4213/tvp2456

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English version:
Theory of Probability and its Applications, 2009, 53:3, 537–547

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Citation: O. Hadjiliadis, V. H. Poor, “On the Best 2-CUSUM Stopping Rule for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model”, Teor. Veroyatnost. i Primenen., 53:3 (2008), 610–622; Theory Probab. Appl., 53:3 (2009), 537–547

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp2456
• https://doi.org/10.4213/tvp2456
• http://mi.mathnet.ru/eng/tvp/v53/i3/p610

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. B. S. Darhovsky, “Change-point detection in random sequence under minimal prior information”, Theory Probab. Appl., 58:3 (2014), 488–493
2. Komaee A., “Quickest Detection of a Random Pulse in White Gaussian Noise”, IEEE Trans. Autom. Control, 59:6 (2014), 1468–1479
3. Moustakides G.V., “Multiple Optimality Properties of the Shewhart Test”, Seq. Anal., 33:3 (2014), 318–344
4. Zhang H., Hadjiliadis O., Schaefer T., Poor H.V., “Quickest Detection in Coupled Systems”, SIAM J. Control Optim., 52:3 (2014), 1567–1596
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