Sequential change detection revisited. (English) Zbl 1133.62063

Moustakides, George V.

Ann. Stat. 36, No. 2, 787-807 (2008).

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.

MSC:

62L10	Sequential statistical analysis
62L15	Optimal stopping in statistics
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

change-point; disorder problem; sequential detection

Citations:

Zbl 0255.62067

References:

[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 · doi:10.1214/aos/1032298296
[2]	Bayraktar, E. and Dayanik, S. (2006). Poisson disorder problem with exponential penalty for delay. Math. Oper. Res. 31 217-233. · Zbl 1278.62132 · doi:10.1287/moor.1060.0190
[3]	Bayraktar, E., Dayanik, S. and Karatzas, I. (2006). Adaptive Poisson disorder problem. Ann. App. Prob. 16 1190-1261. · Zbl 1104.62093 · doi:10.1214/105051606000000312
[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 · doi:10.1524/stnd.21.1.3.20317
[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 · doi:10.1214/aoms/1177693055
[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 · doi:10.1214/009053606000000362
[8]	Moustakides, G. V. (1986). Optimal stopping times for detecting changes in distributions. Ann. Statist. 14 1379-1387. · Zbl 0612.62116 · doi:10.1214/aos/1176350164
[9]	Moustakides, G. V. (2004). Optimality of the CUSUM procedure in continuous time. Ann. Statist. 32 302-315. · Zbl 1105.62368 · doi:10.1214/aos/1079120138
[10]	Page, E. S. (1954). Continuous inspection schemes. Biometrika 41 100-115. JSTOR: · Zbl 0056.38002 · doi:10.1093/biomet/41.1-2.100
[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 · doi:10.1214/aos/1176346587
[13]	Poor, H. V. (1998). Quickest detection with exponential penalty for delay. Ann. Statist. 26 2179-2205. · Zbl 0927.62077 · doi:10.1214/aos/1024691466
[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 · doi:10.1214/aos/1176347761
[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 · doi:10.1070/RM1996v051n04ABEH002986
[19]	Taylor, H. M. (1975). A stopped Brownian Motion formula. Ann. Probab. 3 234-246. JSTOR: · Zbl 0303.60072 · doi:10.1214/aop/1176996395