vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2026-62-3-211-222vestifm-916Research ArticleМАТЕМАТИКАMATHEMATICSСтатистический анализ двоичной цепи Маркова с разладкойStatistical analysis of binary Markov chain with a change pointХаринЮ. С.KharinYu. S.

Харин Юрий Семенович – академик Национальной академии наук Беларуси, доктор физико-математических наук, профессор, директор

пр. Независимости, 4, 220030, Минск

Yuriy S. Kharin – Academician of the National Academy of Sciences of Belarus, D. Sc. (Physics and Mathematics), Professor, Director

4, Nezavisimosti Ave., Minsk, 220030

kharin@bsu.byШибалкоС. А.ShibalkoS. A.

Шибалко Сергей Анатольевич – магистрант, младший научный сотрудник

пр. Независимости, 4, 220030, Минск

Shibalko Siarhei A. – Master Student, Junior Researcher

4, Nezavisimosti Ave., Minsk, 220030

shibalko2003@bk.ruНИИ прикладных проблем математики и информатики Белорусского государственного университетаБеларусьInstitute for Applied Problems of Mathematics and In formatics of the Belarusian State UniversityBelarus202629092026623211222Copyright © Харин Ю.С., Шибалко С.А., 20262026Харин Ю.С., Шибалко С.А.Kharin Y.S., Shibalko S.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/916

Рассматривается задача статистического анализа двоичной неоднородной цепи Маркова с одной разладкой (моментом времени «скачкообразного» изменения матрицы вероятностей одношаговых переходов). Исследованы два случая: при известных матрицах переходных вероятностей и при априорно неизвестных матрицах переходных вероятностей. Построены состоятельные статистические оценки параметров, доказаны их асимптотические свойства при увеличении длительности наблюдения. Разработан алгоритм статистического оценивания момента разладки. Представлены результаты компьютерных экспериментов.

This paper is devoted to statistical analysis of a binary heterogeneous Markov chain with a single change point (time point of the change in the one-step transition probability matrix). To address this problem, two cases are studied: one where the transition probability matrices are known, and another where they are unknown a priori. Consistent statistical estimators for the parameters are constructed, their asymptotic properties are proved, and an algorithm for estimating the change point is developed. The results of computer experiments are presented.

двоичный временной ряднеоднородная цепь Марковамомент разладкистатистическое оценивание параметровструктурные измененияbinary time seriesheterogeneous Markov chainchange pointstatistical parameter estimationstructural changeReferencesХарин, Ю. С. Современные тенденции в компьютерном анализе данных / Ю. С. Харин, С. В. Абламейко // Журнал Белорусского государственного университета. Математика. Информатика. – 2026. – № 1. – С. 6 –15.Kharin Yu. S., Ablameyko S. V. Current trends in computer data analysis. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2026, no. 1, pp. 6–15 (in Russian).Statistical analysis of multivariate discrete-valued time series / K. Fokianos, R. Fried, Yu. Kharin, V. Voloshko // Journal of Multivariate Analysis. – 2022. – Vol. 188. – Art. ID 104805. https://doi.org/10.1016/j.jmva.2021.104805Fokianos K., Fried R., Kharin Yu., Voloshko V. Statistical analysis of multivariate discrete-valued time series. Journal of Multivariate Analysis, 2022, vol. 188, art. ID 104805. https://doi.org/10.1016/j.jmva.2021.104805Kong, J. Poisson Count Time Series / J. Kong, R. Lund // Journal of Time Series Analysis. – 2026. – Vol. 47, № 2. – P. 279–303. https://doi.org/10.1111/jtsa.12799Kong J., Lund R. Poisson Count Time Series. Journal of Time Series Analysis, 2026, vol. 47, pp. 279–303. https://doi.org/10.1111/jtsa.12799.Bastian, P. Multiscale detection of practically significant changes in a gradually varying time series / P. Bastian, H. Dette // Electronic Journal of Statistics. – 2026. – Vol. 20, № 1. – P. 138–162. https://doi.org/10.1214/26-ejs2485Bastian P., Dette H. Multiscale detection of practically significant changes in a gradually varying time series. Electronic Journal of Statistics, 2026, vol. 20, no. 2, pp. 138–162. https://doi.org/10.1111/jtsa.12799Wang, D. Univariate mean change point detection: Penalization, CUSUM and optimality / D. Wang, Y. Yu, A. Rinaldo // Electronic Journal of Statistics. – 2020. – Vol. 14, № 1. – P. 1917–1961. https://doi.org/10.1214/26-ejs2485Wang D., Yu Y., Rinaldo A. Univariate mean change point detection: Penalization, CUSUM and optimality. Electronic Journal of Statistics, 2020, vol. 14, no. 1, pp. 1917–1961. https://doi.org/10.1214/26-ejs2485Kirch, C. Detection of change points in discrete valued time series / C. Kirch, J. Tajduidje Kamgaing // Handbook of Discrete-Valued Time Series. – 2016. – Vol. 8, № 2. – P. 235–255. – https://doi.org/10.1201/b19485-12Kirch C., Tajduidje Kamgaing J. Detection of change points in discrete valued time series. Handbook of Discrete- Valued Time Series, 2016, vol. 8, no. 2, pp. 235–255. https://doi.org/10.1201/b19485-12Berchtold, A. High-Order Extensions of the Double Chain Markov Model / A. Berchtold; University of Washington Department of Statistics. – Seattle, 1999. – (Technical Report; N 356). – URL: https://stat.uw.edu/research/tech-reports/higher-order-extensions-double-chain-markov-modelBerchtold A. High-Order Extensions of the Double Chain Markov Model. Technical Report no. 356. Seattle, 1999. Available at: https://stat.uw.edu/research/tech-reports/higher-order-extensions-double-chain-markov-modelХарин, Ю. С. Неоднородные цепи Маркова высокого порядка: модели и их вероятностно-статистический анализ / Ю. С. Харин // Доклады Национальной академии наук Беларуси. – 2026. – Т. 70, № 3. – С. 191–200. https://doi.org/10.29235/1561-8323-2026-70-3-191-200Kharin Yu. S. Non-homogeneous high-order Markov chains: models and their probabilistic-statistical analysis. Doklady Natsional’noi akademii nauk Belarusi = Doklady of the National Academy of Sciences of Belarus, 2026, vol. 70, no. 3, pp. 191–200 (in Russian). https://doi.org/10.29235/1561-8323-2026-70-3-191-200Billingsley, P. Statistical Methods in Markov Chains / P. Billingsley // The Annals of Mathematical Statistics. 1961. – Vol. 32, № 1. – P. 12–40. https://doi.org/10.1214/aoms/1177705136Billingsley P. Statistical Methods in Markov Chains. The Annals of Mathematical Statistics, 1961, vol. 32, no. 1, pp. 12–40. https://doi.org/10.1214/aoms/1177705136Shiryaev, A. N. Probability / A. N. Shiryaev. – New York: Springer, 1995. – 624 p.Shiryaev A. N. Probability. New York, Springer, 1995. 624 p.Van der Vaart, A. W. Weak Convergence and Empirical Processes: With Applications to Statistics / A. W. Van der Vaart, J. A. Wellner. – New York: Springer, 1996. – XVI, 508 p. https://doi.org/10.1007/978-1-4757-2545-2Van der Vaart A. W., Wellner J. A. Weak Convergence and Empirical Processes: With Applications to Statistics. New York, Springer, 1996. XVI, 508 p. https://doi.org/10.1007/978-1-4757-2545-2

The authors declare that there are no conflicts of interest present.