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Optimality of the 2-CUSUM drift equalizer rules for detecting two-sided alternatives in the Brownian motion model

Part of: Game theory Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Olympia Hadjiliadis
Olympia Hadjiliadis*
Affiliation:
Columbia University
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Abstract

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This work employs the Brownian motion model in which observations are taken sequentially. The objective is to detect a two-sided change in the constant drift by means of a stopping rule. As a performance measure, an extended Lorden criterion is used. The goal is to minimize the worst-case detection delay subject to a constraint in the frequency of false alarms. In a companion paper, attention is drawn to a first category of 2-CUSUM rules for which the harmonic mean rule holds. It is further seen that a special class of 2-CUSUM stopping rules within this category, called drift equalizer rules, perform strictly better than non-equalizer rules, according to this specific performance measure.

Keywords

Sequential change detectiondetection delayCUSUMtwo-sided CUSUM

MSC classification

Primary: 62L15: Optimal stopping
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 91A60: Probabilistic games; gambling
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1183 - 1193
Copyright
© Applied Probability Trust 2005 

References

Bakut, P. et al. (1963). Statistical Radar Theory, Vol. 1, 1st edn. Soviet Radio, Moscow (in Russian).Google Scholar
Barnard, G. (1959). Control charts and stochastic processes. J. R. Statist. Soc. B 11, 239–271.Google Scholar
Basseville, M. and Nikiforov, I. (1993). Detection of Abrupt Changes: Theory and Applications. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
Beibel, M. (1996). A note on Ritov's Bayes approach to the minimax property of the cusum procedure. Ann. Statist. 24, 1804–1812.CrossRefGoogle Scholar
Bussgang, J. (1970). Sequential methods in radar detection. Proc. IEEE 58, 731–743.CrossRefGoogle Scholar
Dragalin, V. (1994). Optimality of a generalized CUSUM procedure in quickest detection problem. In Statistics and Control of Random Processes (Proc. Steklov Inst. Math. 202), American Mathematical Society, Providence, RI, pp. 107–120.Google Scholar
Dragalin, V. (1997). The design and analysis of 2-CUSUM} procedure. Commun. Statist. Simul. Comput. 26, 67–81.CrossRefGoogle Scholar
Frisen, M. (1992). Evaluations of methods for statistical surveillance. Statist. Med. 11, 1489–1502.CrossRefGoogle ScholarPubMed
Fu, K. (1968). Sequential Methods in Pattern Recognition and Learning, 1st edn. Academic Press, New York.Google Scholar
Hadjiliadis, O. and Moustakides, G. (2005). Optimal and asymptotically optimal CUSUM rules for change point detection in the Brownian motion model with multiple alternatives. Teor. Veroyat. Primen. 50, 131–144. English translation: to appear in Theory Prob. Appl. 50.CrossRefGoogle Scholar
Long, M. (1992). Airborne Early Warning System Concepts, 1st edn. Artech, Boston, MA.Google Scholar
Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Statist. 42, 1897–1908.CrossRefGoogle Scholar
Moustakides, G. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14, 1379–1387.CrossRefGoogle Scholar
Moustakides, G. (2004). Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32, 302–315.CrossRefGoogle Scholar
Page, E. (1954). Continuous inspection schemes. Biometrika 41, 100–115.CrossRefGoogle Scholar
Pollak, M. and Siegmund, D. (1985). A diffusion process and its applications to detecting a change in the drift of Brownian motion. Biometrika 72, 267–280.CrossRefGoogle Scholar
Poor, V. (1998). Quickest detection with exponential penalty for delay. Ann. Statist. 26, 2179–2205.CrossRefGoogle Scholar
Radaelli, G. (1992). Using the cuscore technique in the surveillance of rare health events. J. Appl. Statist. 19, 75–81.CrossRefGoogle Scholar
Roberts, S. (1959). Control chart tests based on geometric moving averages. Technometrics 1, 239–250.CrossRefGoogle Scholar
Roberts, S. (1966). A comparison of some control chart procedures. Technometrics 8, 411–430.CrossRefGoogle Scholar
Shiryaev, A. N. (1963). On optimum methods in quickest detection problems. Theory Prob. Appl. 13, 22–46.CrossRefGoogle Scholar
Shiryaev, A. N. (1996). Minimax optimality of the method of cumulative sums ({CUSUM}) in the case of continuous time. Russian Math. Surveys 51, 750–751.CrossRefGoogle Scholar
Siegmund, D. (1985). Sequential Analysis, 1st edn. Springer, New York.CrossRefGoogle Scholar
Sonesson, C. (2003). Evaluations of some exponentially weighted moving average methods. J. Appl. Statist. 30, 1115–1133.CrossRefGoogle Scholar
Srivastava, M. and Wu, Y. (1993). Comparison of EWMA, CUSUM and Shiryayev-Roberts procedures for detecting a shift in the mean. Ann. Statist. 21, 645–670.CrossRefGoogle Scholar
Tartakovsky, A. (1994). Asymptotically minimax multi-alternative sequential rule for disorder detection. In Statistics and Control of Random Processes (Proc. Steklov Inst. Math. 202), American Mathematical Society, Providence, RI, pp. 229–236.Google Scholar
Tartakovsky, A. (1995). Asymptotic properties of CUSUM and Shiryaev's procedures for detecting a change in non-homogeneous Gaussian processes. Math. Meth. Statist. 4, 389–404.Google Scholar