On Stochastic Models and Optimal Methods in the Quickest Detection Problems

A. N. Shiryaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 3, pp. 417-436

Résumé

This paper is an introduction to the thematic issue devoted to the optimal and asymptotically optimal methods of decision making in problems of the quickest detection of changes of probability characteristics of observed processes (the “disorder” problem), as well as some general problems of the optimal stopping theory on which the decision of these problems is based. This paper's introductory purpose is twofold: on the one hand it gives a general model covering a variate of schemes describing the appearance of disorder, and on the other hand it describes briefly the specific models and general problems concerning the optimal stopping theory which the papers of this issue contain.

Détail

Keywords: control charts procedure, CUSUM-method, quickest detection problem, $\theta$-model, Bayesian $G$-model, stopping times, optimal stopping rules.

Bibliographie

[1] Shewhart W. A., “The application of statistics as an aid in maintaining quality of a manufactured product”, J. Amer. Statist. Assoc., 20:152 (1925), 546–548 | DOI

[2] Shewhart W. A., Economic Control of Quality of Manufactured Product, Van Nostrand Rainhold, New York, 1931, Republished in 1980 by the American Society for Quality Control, Milwaukee

[3] Page E. S., “Continuous inspection schemes”, Biometrika, 41 (1954), 100–114 | MR

[4] Page E. S., “Control charts with warning lines”, Biometrika, 42 (1955), 243–257 | MR | Zbl

[5] Roberts S. W., “Control charts based on geometric moving average”, Technometrics, 1 (1959), 239–250 | DOI

[6] Roberts S. W., “A comparison of some control chart procedures”, Technometrics, 8 (1966), 411–430 | DOI | MR

[7] Wetherill G. B., Sampling Inspection and Quality Control, Chapman, London; Halsted Press, New York, 1977, 146 pp. | MR | Zbl

[8] Hawkins D. M., Olwell D. H., Cumulative Sum Charts and Charting for Quality Improvement, Springer-Verlag, New York, 1998, 247 pp. | MR

[9] Shiryaev A. N., “Obnaruzhenie spontanno voznikayuschikh effektov”, Dokl. AN SSSR, 138:4 (1961), 799–801

[10] Shiryaev A. N., “Zadacha skoreishego obnaruzheniya narusheniya statsionarnogo rezhima”, Dokl. AN SSSR, 138:5 (1961), 1039–1042 | MR | Zbl

[11] Shiryaev A. N., “Ob optimalnykh metodakh v zadachakh skoreishego obnaruzheniya”, Teoriya veroyatn. i ee primen., 8:1 (1963), 26–51 | Zbl

[12] Shiryaev A. N., “K obnaruzheniyu razladok proizvodstvennogo protsessa. I, II”, Teoriya veroyatn. i ee primen., 8:3 (1963), 264–281 ; 4, 431–445

[13] Kolmogorov A. N., Prokhorov Yu. V., Shiryaev A. N., “Veroyatnostno-statisticheskie metody obnaruzheniya spontanno voznikayuschikh effektov”, Tr. MIAN, 182, 1988, 4–23 | Zbl

[14] Shiryaev A. N., Statisticheskii posledovatelnyi analiz, 2-e izd., Nauka, M., 1976, 272 pp.

[15] “Roundtable on bioterrorism detection: Information system-based surveillance”, J. Amer. Medical Informatics Assoc., 9 (2002), 105–115 | DOI

[16] Frisén M., “Properties of use of the Shewhart method and its followers”, Sequential Anal., 26:2 (2007), 171–193 | DOI | MR | Zbl

[17] Tartakovsky A. G., Veeravalli V., “Change-point detection in multichannel and distributed systems”, Applied Sequential Methodologies, eds. M. Mukhopadhyay, S. Datta, and S. Chattopadhaya, Dekker, New York, 2004, 339–370 | MR | Zbl

[18] Sequential Analysis. Design Methods and Applications, 26:2–3 (2007), Celebrating eighty years of control charts. Walter A. Shewhart's Legacy. Taylor and Francis

[19] Kavtaradze T., Lazrieva N., Mania M., Muliere P., “A Bayesian-martingale approach to the general disorder problem”, Stochastic Process. Appl., 117:8 (2007), 1093–1120 | DOI | MR | Zbl

[20] Jacod J., Shiryaev A. N., Limit Theorem for Stochastic Processes, 2nd ed., Springer-Verlag, Berlin, 2003, 661 pp. | MR

[21] Shiryaev A. N., Veroyatnost, 4-e izd., MTsNMO, M., 2007, 967 pp.

[22] Bojdecki T., “Probability maximizing approach to optimal stopping and its application to a disorder problem”, Stochastics, 3:1 (1979), 61–71 | MR | Zbl

[23] Shiryaev A. N., “Quickest detection problems in the technical analysis of the financial data”, Mathematical Finance–Bachelier Congress 2000: Selected papers from the First World Congress of the Bachelier Finance Society (Paris, 2000), Springer Finance, eds. H. Geman et al., Springer-Verlag, Berlin, 2002, 487–521 | MR | Zbl

[24] Shiryaev A. N., “Ot “razladki” k nelineinoi filtratsii i teorii martingalov”, Matematicheckie sobytiya XX veka, FAZIS, M., 2003, 491–518

[25] Peskir G., Shiryaev A. N., Optimal Stopping and Free-Boundary Problems, Birkhäuser, Basel, 2006, 500 pp. | MR | Zbl

[26] Feinberg E. A., Shiryaev A. N., “Quickest detection of drift change for Brownian motion in generalized Bayesian and minimax settings”, Statist. Decisions, 24:4 (2006), 445–470 | DOI | MR | Zbl

[27] Moustakides G. V., “Optimal stopping times for detecting changes in distributions”, Ann. Statist., 14:4 (1986), 1379–1387 | DOI | MR | Zbl

[28] Lorden G., “Procedures for reacting to a change in distribution”, Ann. Math. Statist., 42 (1971), 1897–1908 | DOI | MR | Zbl

[29] Pollak M., “Optimal detection of a change in distribution”, Ann. Statist., 13:1 (1985), 206–227 | DOI | MR | Zbl

[30] Pollak M., Siegmund D., “A diffusion process and its application to detecting a change in the drift of Brownian motion”, Biometrika, 72:2 (1985), 267–280 | DOI | MR | Zbl

[31] Beibel M., “A note on Ritov's Bayes approach to the minimax property of the CUSUM procedure”, Ann. Statist., 24:4 (1996), 1804–1812 | DOI | MR | Zbl

[32] Shiryaev A. N., “Minimaksnaya optimalnost metoda kumulyativnykh summ (CUSUM) v sluchae nepreryvnogo vremeni”, Uspekhi matem. nauk, 51:4 (1996), 173–174 | MR | Zbl