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Sequential change detection revisited. (English) Zbl 1133.62063
Summary: In sequential change detection, existing performance measures differ significantly in the way they treat the time of change. By modeling this quantity as a random time, we introduce a general framework capable of capturing and better understanding most well-known criteria and also propose new ones. For a specific new criterion that constitutes an extension of G. Lorden’s [Ann. Math. Stat. 42, 1897–1908 (1971; Zbl 0255.62067)] performance measure, we offer the optimum structure for detecting a change in the constant drift of a Brownian motion and a formula for the corresponding optimum performance.

62L10 Sequential statistical analysis
62L15 Optimal stopping in statistics
60G40 Stopping times; optimal stopping problems; gambling theory
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[1] Beibel, M. (1996). A note on Ritov’s Bayes approach to the minmax property of the CUSUM procedure. Ann. Stat. 24 1804-1812. · Zbl 0868.62063
[2] Bayraktar, E. and Dayanik, S. (2006). Poisson disorder problem with exponential penalty for delay. Math. Oper. Res. 31 217-233. · Zbl 1278.62132
[3] Bayraktar, E., Dayanik, S. and Karatzas, I. (2006). Adaptive Poisson disorder problem. Ann. App. Prob. 16 1190-1261. · Zbl 1104.62093
[4] Karatzas, I. (2003). A note on Bayesian detection of change-points with an expected miss criterion. Stat. Decis. 21 3-13. · Zbl 1037.62080
[5] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus . Springer, New York. · Zbl 0638.60065
[6] Lorden, G. (1971). Procedures for reacting to a change in distribution. Ann. Math. Stat. 42 1897-1908. · Zbl 0255.62067
[7] Mei, Y. (2006). Comment on “A note on optimal detection of a change in distribution” by Benjamin Yakir. Ann. Statist. 34 1570-1576. · Zbl 1113.62092
[8] Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379-1387. · Zbl 0612.62116
[9] Moustakides, G. V. (2004). Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32 302-315. · Zbl 1105.62368
[10] Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100-115. JSTOR: · Zbl 0056.38002
[11] Peskir, G. and Shiryaev, A. N. (2002). Solving the Poisson disorder problem. In Advances in Finance and Stochastics (K. Sandmann and P. J. Schönbucher, eds.) 295-312. Springer, Berlin. · Zbl 1009.60033
[12] Pollak, M. (1985). Optimal detection of a change in distribution. Ann. Statist. 13 206-227. · Zbl 0573.62074
[13] Poor, H. V. (1998). Quickest detection with exponential penalty for delay. Ann. Statist. 26 2179-2205. · Zbl 0927.62077
[14] Protter, P. E. (2004). Stochastic Integration and Differential Equations , 2nd ed. Springer, Berlin. · Zbl 1041.60005
[15] Ritov, Y. (1990). Decision theoretic optimality of the CUSUM procedure. Ann. Statist. 18 1464-1469. · Zbl 0712.62073
[16] Rodionov, S. R. and Overland, J. E. (2005). Application of a sequential regime shift detection method to the Bering Sea ecosystem. ICES J. Marine Science 62 328-332.
[17] Shiryayev, A. N. (1978). Optimal Stopping Rules . Springer, New York. · Zbl 0391.60002
[18] Shiryayev, A. N. (1996). Minimax optimality of the method of cumulative sums (CUSUM) in the case of continuous time. Russ. Math. Surv. 51 750-751. · Zbl 0882.62076
[19] Taylor, H. M. (1975). A stopped Brownian Motion formula. Ann. Probab. 3 234-246. JSTOR: · Zbl 0303.60072
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