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"10.29235/1561-2430-2022-58-3-280-291vestifm-664Research ArticleМАТЕМАТИКАMATHEMATICSДискретные временные ряды на основе экспоненциального семейства с многомерным параметром и их вероятностно-статистический анализDiscrete-valued time series based on the exponential family with the multidimensional parameter and their probabilistic and statistical analysis.ВолошкоВ. А.VoloshkoV. A.

Волошко Валерий Анатольевич – кандидат физико-математических наук, заведующий сектором компьютерного анализа данных

пр. Независимости, 4, 220030, Минск

Valeriy A. Voloshko – Ph. D. (Physics and Mathematics), Head of the Computer Data Analysis Sector

Nezavisimosty Ave., 4, Minsk, 220030

valeravoloshko@yandex.ruХаринЮ. С.KharinYu. S.

Харин Юрий Семенович – академик Национальнойакадемии 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, Dr. Sc. (Physics and Mathematics), Professor

Nezavisimosty Ave., 4, Minsk, 220030

kharin@bsu.byНаучно-исследовательский институт прикладных проблем математики и информатики, Белорусский государственный университетResearch Institute for Applied Problems of Mathematics and Informatics, Belarusian State University202212102022583280291Copyright © Волошко В.А., Харин Ю.С., 20222022Волошко В.А., Харин Ю.С.Voloshko V.A., Kharin Y.S.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/664

Предложена новая малопараметрическая модель дискретного временного ряда на основе экспоненциального семейства дискретных распределений вероятностей с многомерным параметром. Для параметров предложенной модели строится семейство состоятельных асимптотически нормальных статистических оценок явного вида, в котором найдена асимптотически эффективная оценка, достигающая границы Крамера – Рао при растущей длительности наблюдения временного ряда. Полученные результаты могут быть использованы для робастного статистического анализа дискретных временных рядов, статистического анализа дискретных пространственно-временных данных и случайных полей.

We propose herein a new parsimonious Markov model for a discrete-valued time series with conditional probability distributions of observations lying in the exponential family with the multidimensional parameter. A family of explicit consistent asymptotically normal statistical estimators is constructed for the parameters of the proposed model for increasing length of observed time series, and asymptotically effective estimator is found within this constructed family. The obtained results can be used for robust statistical analysis of discrete-valued time series,and for statistical analysis of discrete-valued spatio-temporal data and random fields.

дискретный временной рядмалопараметрическая модельцепь Маркова высокого порядкаэкспоненциальное семействоэффективная оценкаdiscrete-valued time seriesparsimonious modelhigh-order Markov chainexponential familyeffective estimatorReferencesStatistical analysis of multivariate discrete-valued time series / K. Fokianos [et al.] // J. Multivar. Anal. – 2022. – Vol. 188. – P. 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, pp. 104805. https://doi.org/10.1016/j.jmva.2021.104805Kharin, Yu. Statistical analysis of discrete-valued time series by parsimonious high-order Markov chains / Yu. Kharin // Austr J. Statistics. – 2020. – Vol. 49, № 4. – P. 76–88. https://doi.org/10.17713/ajs.v49i4.1132Kharin Yu. Statistical analysis of discrete-valued time series by parsimonious high-order Markov chains. Austrian Journal of Statistics, 2020, vol. 49, no. 4, pp. 76–88. https://doi.org/10.17713/ajs.v49i4.1132Anderson, T. W. The Statistical Analysis of Time Series / T. W. Anderson. – New York: Wiley, 1971. https://doi. org/10.1002/9781118186428Anderson T. W. The Statistical Analysis of Time Series. New York, Wiley, 1971. https://doi.org/10.1002/9781118186428Box, G. Time Series Analysis: Forecasting and Control / G. Box, G. Jenkins. – San Francisco: Holden-Day, 1970.Box G., Jenkins G. Time Series Analysis: Forecasting and Control. San Francisco, Holden-Day, 1970.Nelder, J. Generalized linear models / J. Nelder, R. Wedderburn // J. R. Stat. Soc., Ser. A. – 1972. – Vol. 135, № 3. – P. 370–384. https://doi.org/10.2307/2344614Nelder J., Wedderburn R. Generalized linear models. Journal of the Royal Statistical Society. Series A, 1972, vol. 135, no. 3, pp. 370–384. https://doi.org/10.2307/2344614Kedem, B. Regression Models for Time Series Analysis / B. Kedem, K. Fokianos. – Hoboken: Wiley, 2002. https://doi. org/10.1002/0471266981Kedem B., Fokianos K. Regression Models for Time Series Analysis. Hoboken, Wiley, 2002. https://doi. org/10.1002/0471266981Amari, S. Methods of Information Geometry / S. Amari, H. Nagaoka. – Oxford University Press, 2000. https://doi. org/10.1090/mmono/191Amari S., Nagaoka H. Methods of Information Geometry. Oxford University Press, 2000. https://doi.org/10.1090/mmono/191Yee, T. W. Vector Generalized Linear and Additive Models: With an Implementation in R / T. W. Yee. – New York: Springer, 2015. https://doi.org/10.1007/978-1-4939-2818-7Yee T. W. Vector Generalized Linear and Additive Models: With an Implementation in R. New York, Springer, 2015. https://doi.org/10.1007/978-1-4939-2818-7Kharin, Yu. Robust estimation for binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies / Yu. Kharin, V. Voloshko // J. Multivar. Anal. – 2021. – Vol. 185. – P. 104777. https://doi.org/10.1016/j. jmva.2021.104777Kharin Yu., Voloshko V. Robust estimation for binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies. Journal of Multivariate Analysis, 2021, vol. 185, pp. 104777. https://doi.org/10.1016/j. jmva.2021.104777Харин, Ю. С. Семибиномиальные условно нелинейные авторегрессионные модели дискретных случайных последовательностей: вероятностные свойства и статистическое оценивание параметров / Ю. С. Харин, В. А. Волошко // Дискрет. математика. – 2019. – Вып. 31, № 1. – С. 72–98.Kharin Yu. S., Voloshko V. A. Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation. Discrete Mathematics and Applications, 2020, vol. 30, no. 6, pp. 417–437. https://doi.org/10.1515/dma-2020-0038Харин, Ю. С. Биномиальная условно нелинейная авторегрессионная модель дискретного временного ряда и ее вероятностно-статистические свойства / Ю. С. Харин, В. А. Волошко // Тр. Ин-та математики. – 2019. – Вып. 26, № 1. – С. 95–105.Kharin Yu., Voloshko V. Binomial conditionally nonlinear autoregressive model of discrete-valued time series and its probabilistic and statistical properties. Trudy Instituta matematiki = Proceedings of the Institute of Mathematics, 2019, vol. 26, no. 1, pp. 95–105 (in Russian).Kharin, Yu. S. Statistical estimation of parameters for binary conditionally nonlinear autoregressive time series / Yu. S. Kharin, V. A. Voloshko, E. A. Medved // Math. Methods Stat. – 2018. – Vol. 27, № 2. – P. 103–118. https://doi. org/10.3103/s1066530718020023Kharin Yu. S., Voloshko V. A., Medved E. A. Statistical estimation of parameters for binary conditionally nonlinear autoregressive time series. Mathematical Methods of Statistics, 2018, vol. 27, no. 2, pp. 103–118. https://doi.org/10.3103/ s1066530718020023Andersen, E. B. Sufficiency and exponential families for discrete sample spaces / E. B. Andersen // J. Am. Stat. Assoc. – 1970. – Vol. 65, № 331. – P. 1248–1255. https://doi.org/10.1080/01621459.1970.10481160Andersen E. B. Sufficiency and exponential families for discrete sample spaces. Journal of the American Statistical Association, 1970, vol. 65, no. 331, pp. 1248–1255. https://doi.org/10.1080/01621459.1970.10481160Kharin, Yu. S. Statistical analysis of conditionally binomial nonlinear regression time series with discrete regressors / Yu. S. Kharin, V. A. Voloshko // Theory Probab. Math. Stat. – 2020. – Vol. 100. – P. 181–190. https://doi.org/10.1090/tpms/1105Kharin Yu. S., Voloshko V. A. Statistical analysis of conditionally binomial nonlinear regression time series with discrete regressors. Theory of Probability and Mathematical Statistics, 2020, vol. 100, pp. 181–190. https://doi.org/10.1090/ tpms/1105Engel, J. Polytomous logistic regression / J. Engel // Stat. Neerlandica. – 1988. – Vol. 42, № 4. – P. 233–252. https://doi. org/10.1111/j.1467-9574.1988.tb01238.xEngel J. Polytomous logistic regression. Statistica Neerlandica, 1988, vol. 42, no. 4, pp. 233–252. https://doi. org/10.1111/j.1467-9574.1988.tb01238.xKrisztin, T. A spatial multinomial logit model for analysing urban expansion / T. Krisztin, P. Piribauer, M. Wögerer // Spat. Econ. Anal. – 2022. – Vol. 17, № 2. – P. 223–244. https://doi.org/10.1080/17421772.2021.1933579Krisztin T., Piribauer P., Wögerer M. A spatial multinomial logit model for analysing urban expansion. Spatial Economic Analysis, 2022, vol. 17, no. 2, pp. 223–244. https://doi.org/10.1080/17421772.2021.1933579Billingsley, P. Statistical methods in Markov chains / P. Billingsley // Ann. Math. Stat. – 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/1177705136Kharin, Yu. Robustness in statistical pattern recognition under “contaminations” of training samples / Yu. Kharin, E. Zhuk // Proc. of the 12th IAPR Int. Conf. on Pattern Recognition. – 1994. – Vol. 2. – P. 504–506. https://doi.org/10.1109/ icpr.1994.576996Kharin Yu., Zhuk E. Robustness in statistical pattern recognition under “contaminations” of training samples. Proceedings of the 12th IAPR International Conference on Pattern Recognition, 1994, vol. 2, pp. 504–506. https://doi. org/10.1109/icpr.1994.576996Maevskii, V. V. Robust regressive forecasting under functional distortions in a model / V. V. Maevskii, Yu. S. Kharin // Automation and Remote Control. – 2002. – Vol. 63, № 11. – P. 1803–1820. https://doi.org/10.1023/a:1020959432568Maevskii V. V., Kharin Yu. S. Robust regressive forecasting under functional distortions in a model. Automation and Remote Control, 2002, vol. 63, no. 11, pp. 1803–1820. https://doi.org/10.1023/a:1020959432568Kharin, Yu. Statistical analysis of spatio-temporal data based on Poisson conditional autoregressive model / Yu. Kharin, M. Zhurak // Informatica. – 2015. – Vol. 26, № 1. – P. 67–87. https://doi.org/10.15388/informatica.2015.39Kharin Yu., Zhurak M. Statistical analysis of spatio-temporal data based on Poisson conditional autoregressive model. Informatica, 2015, vol. 26, no. 1, pp. 67–87. https://doi.org/10.15388/informatica.2015.39

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