On a random number of disorders

• Autorzy: Szajkowski K.
• Języki publikacji: EN

Abstrakty

EN

We register a random sequence which has three segments being the homogeneous Markov processes. Each segment has its own onestep transition probability law and the length of the segment is unknown and random. It means that at two random moments θ₁, θ₂, where 0 ≤ θ₁ ≤ θ₂, the source of observation is changed. In effect, the number of homogeneous segments is random. The transition probabilities of each process are known and the a priori distribution of the disorder moments is given. The former research on such a problem has been devoted to various questions concerning the distribution changes. The random number of distributional segments creates new problems in solutions with relation to analysis of the model with deterministic number of segments. Two cases are presented in detail. In the first one the objective is to stop on or between the disorder moments while in the second one our objective is to find the strategy which immediately detects the distribution changes. Both problems are reformulated to optimal stopping of the observed sequences. The detailed analysis of the problem is presented to show the form of optimal decision function.

Słowa kluczowe

EN

disorder problem; sequential detection; optimal stopping; Markov process; change point; double optimal stopping

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 1
• Strony: 17--45
• Opis fizyczny: Bibliogr. 25 poz.

Twórcy

autor

Szajkowski K.

• Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

Bibliografia

• [1] M. Baron, Early detection of epidemics as a sequential change-point problem, in: Longevity, Aging and Degradation Models in Reliability, Public Health, Medicine and Biology, LAD 2004. Selected Papers from the First French-Russian Conference, St. Petersburg, Russia, June 7-9, 2004, V. Antonov, C. Huber, M. Nikulin and V. Polischook (Eds.), IMS Lecture Notes Monogr. Ser., Vol. 2, St. Petersburg State Politechnical University, St. Petersburg, Russia, 2004, pp. 31-43.
• [2] M. Basseville and A. Benveniste, Detection of Abrupt Changes in Signals and Dynamical Systems, Lecture Notes in Control and Inform. Sci., Vol. 77, Springer, Berlin 1986, p. 373.
• [3] T. Bojdecki, Probability maximizing approach to optimal stopping and its application to a disorder problem, Stochastics 3 (1979), pp. 61-71.
• [4] T. Bojdecki and J. Hosza, On a generalized disorder problem, Stochastic Process. Appl. 18 (1984), pp. 349-359.
• [5] B. E. Brodsky and B. S. Darkhovsky, Nonparametric Methods in Change-Point Problems, Math. Appl. (Dordrecht) 243, Kluwer Academic Publishers, 1993, p. 224.
• [6] P. Dube and R. Mazumdar, A framework for quickest detection of traffic anomalies in networks, Technical report, Electrical and Computer Engineering, Purdue University, November 2001; http://citeseer.ist.psu.edu/506551.html.
• [7] Ch.-D. Fuh, Asymptotic operating characteristics of an optimal change point detection in hidden Markov models, Ann. Statist. 32 (5) (2004), pp. 2305-2339.
• [8] G. W. Haggstrom, Optimal sequential procedures when more than one stop is required, Ann. Math. Statist. 38 (1967), pp. 1618-1626.
• [9] T. L. Lai, Sequential changepoint detection in quality control and dynamical systems (with discussion), J. R. Statist. Soc., Ser. B 57 (4) (1995), pp. 613-658.
• [10] T. L. Lai, Information bounds and quick detection of parameter changes in stochastic systems, IEEE Trans. Inform. Theory 44 (7) (1998), pp. 2917-2929.
• [11] G. V. Moustakides, Quickest detection of abrupt changes for a class of random processes, IEEE Trans. Inform. Theory 44 (5) (1998), pp. 1965-1968.
• [12] M. L. Nikolaev, Extended sequential procedures (in Russian), Litovsk. Mat. Sb. 19 (1979), pp. 35-44.
• [13] M. L. Nikolaev, On the criterion of optimality of the extended sequential procedure (in Russian), Litovsk. Mat. Sb. 21 (1981), pp. 75-82.
• [14] L. Pelkowitz, The general discrete-time disorder problem, Stochastics 20 (1987), pp. 89-110.
• [15] L. Pelkowitz, The general Markov chain disorder problem, Stochastics 21 (1987), pp. 113-130.
• [16] W. Sarnowski and K. Szajowski, On-line detection of a part of a sequence with unspecified distribution, Statist. Probab. Lett. 78 (15) (2008), pp. 2511-2516.
• [17] W. A. Shewhart, Economic Control of Quality of Manufactured Products, D. Van Nostrand, New York 1931.
• [18] A. N. Shiryaev, The detection of spontaneous effects, Soviet. Math. Dokl. 2 (1961), pp. 740-743. Translation from: Dokl. Akad. Nauk SSSR 138 (1961), pp. 799-801.
• [19] A. N. Shiryaev, Optimal Stopping Rules, Springer, New York-Heidelberg-Berlin 1978.
• [20] K. Szajowski, Optimal on-line detection of outside observation, J. Statist. Plann. Inference 30 (1992), pp. 413-426.
• [21] K. Szajowski, A two-disorder detection problem, Appl. Math. 24 (2) (1996), pp. 231-241.
• [22] A. G. Tartakovsky, B. L. Rozovskii, R. B. Blažek and H. Kim, Detection of intrusions in information systems by sequential change-point methods, Stat. Methodol. 3 (3) (2006), pp. 252-293.
• [23] A. G. Tartakovsky and V. V. Veeravalli, Asymptotically optimal quickest change detection in distributed sensor systems, Sequential Anal. 27 (4) (2008), pp. 441-475.
• [24] B. Yakir, Optimal detection of a change in distribution when the observations form a Markov chain with a finite state space, in: Change-point Problems. Papers from the AMS-IMS-SIAM Summer Research Conference held at Mt. Holyoke College, South Hadley, MA, USA, July 11-16, 1992, E. Carlstein, H.-G. Müller and D. Siegmund (Eds.), IMS Lecture Notes Monogr. Ser., Institute of Mathematical Statistics, Hayward, California, 1994, pp. 346-358.
• [25] M. Yoshida, Probability maximizing approach for a quickest detection problem with complicated Markov chain, J. Inf. Optim. Sci. 4 (1983), pp. 127-145.