Improved model selection method for an adaptive estimation in semimartingale regression models | Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika – Tomsk State University Journal of Mathematics and Mechanics. 2019. № 58. DOI: 10.17223/19988621/58/2

Improved model selection method for an adaptive estimation in semimartingale regression models

This paper considers the problem of robust adaptive efficient estimating of a periodic function in a continuous time regression model with the dependent noises given by a general square integrable semimartingale with a conditionally Gaussian distribution. An example of such noise is the non-Gaussian Ornstein-Uhlenbeck-Levy processes. An adaptive model selection procedure, based on the improved weighted least square estimates, is proposed. Under some conditions on the noise distribution, sharp oracle inequality for the robust risk has been proved and the robust efficiency of the model selection procedure has been established. The numerical analysis results are given.

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Keywords

improved non-asymptotic estimation, least squares estimates, robust quadratic risk, non-parametric regression, semimartingale noise, Ornstein-Uhlenbeck-Levy process, model selection, sharp oracle inequality, asymptotic efficiency, улучшенное неасимптотическое оценивание, оценки наименьших квадратов, робастный квадратический риск, непараметрическая регрессия, семимартин-гальный шум, процесс Орнштейна - Уленбека - Леви, выбор модели, точное оракульное неравенство, асимптотическая эффективность

Authors

Name	Organization	E-mail
Pchelintsev Evgeny A.	Tomsk State University	evgen-pch@yandex.ru
Pergamenshchikov Serguei M.	University of Rouen Normandy	serge.pergamenchtchikov@univ-rouen.fr

Всего: 2

References

Barndorff-Nielsen O.E., Shephard N. Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial mathematics // J. Royal Stat. Soc. 2001. B 63. P. 167-241.

Bertoin J. Levy Processes. Cambridge: Cambridge University Press, 1996.

Cont R., Tankov P. Financial Modelling with Jump Processes. London: Chapman & Hall, 2004.

Fourdrinier D., Pergamenshchikov S. Improved selection model method for the regression with dependent noise // Annals of the Institute of Statistical Mathematics. 2007. V. 59(3). P. 435-464.

Galtchouk L., Pergamenshchikov S. Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models // Journal of Nonparametric Statistics. 2009. V. 21(1). P. 1-16.

Galtchouk L., Pergamenshchikov S. Adaptive asymptotically efficient estimation in hetero-scedastic nonparametric regression // Journal of Korean Statistical Society. 2009. V. 38(4). P. 305-322.

James W., Stein C. Estimation with quadratic loss // Proceedings of the Fourth Berkeley Symposium Mathematics, Statistics and Probability. Berkeley: University of California Press, 1961. V. 1. P. 361-380.

Konev V.V., Pergamenshchikov S.M. Nonparametric estimation in a semimartingale regression model. Part 1. Oracle Inequalities // Tomsk State University Journal of Mathematics and Mechanics. 2009. No. 3(7). P. 23-41.

Konev V.V., Pergamenshchikov S.M. Nonparametric estimation in a semimartingale regression model. Part 2. Robust asymptotic efficiency // Tomsk State University Journal of Mathematics and Mechanics. 2009. No. 4(8). P. 31-45.

Konev V.V., Pergamenshchikov S.M. General model selection estimation of a periodic regression with a Gaussian noise // Annals of the Institute of Statistical Mathematics. 2010. V. 62. P. 1083-1111.

Konev V.V., Pergamenshchikov S.M. Efficient robust nonparametric in a semimartingale regression model // Annals of the Institute of Henri Poincare. Probab. and Stat. 2012. V. 48(4). P. 1217-1244.

Konev V.V., Pergamenshchikov S.M. Robust model selection for a semimartingale continuous time regression from discrete data // Stochastic processes and their applications. 2015. V. 125. P. 294-326.

Konev V., Pergamenshchikov S. and Pchelintsev E. Estimation of a regression with the pulse type noise from discrete data // Theory Probab. Appl. 2014. V. 58(3). P. 442-457.

Pchelintsev E. Improved estimation in a non-Gaussian parametric regression // Stat. Inference Stoch. Process. 2013. V. 16(1). P. 15-28.

Pchelintsev E., Pergamenshchikov S. Oracle inequalities for the stochastic differential equations // Stat. Inference Stoch. Process. 2018. V. 21 (2). P. 469-483.

Pchelintsev E.A., Pchelintsev V.A., Pergamenshchikov S.M. Non asymptotic sharp oracle inequality for the improved model selection procedures for the adaptive nonparametric signal estimation problem // Communications - Scientific Letters of the University of Zilina. 2018. V. 20 (1). P. 72-76.

Pchelintsev E., Pergamenshchikov S. Adaptive model selection method for a conditionally Gaussian semimartingale regression in continuous time. 2018. P. 1-50. Preprint: http://arxiv. org/abs/1811.05319.

Povzun M.A., Pchelintsev E.A. Estimating parameters in a regression model with dependent noises // Tomsk State University Journal of Mathematics and Mechanics. 2017. No. 49. P. 43-51.