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General model selection estimation of a periodic regression with a Gaussian noise. (English) Zbl 1432.62075

Summary: This paper considers the problem of estimating a periodic function in a continuous time regression model with an additive stationary Gaussian noise having unknown correlation function. A general model selection procedure on the basis of arbitrary projective estimates, which does not need the knowledge of the noise correlation function, is proposed. A non-asymptotic upper bound for \(\mathcal{L}_2\)-risk (oracle inequality) has been derived under mild conditions on the noise. For the Ornstein-Uhlenbeck noise the risk upper bound is shown to be uniform in the nuisance parameter. In the case of Gaussian white noise the constructed procedure has some advantages as compared with the procedure based on the least squares estimates (LSE). The asymptotic minimaxity of the estimates has been proved. The proposed model selection scheme is extended also to the estimation problem based on the discrete data applicably to the situation when high frequency sampling can not be provided.

MSC:

62G05 Nonparametric estimation
62G08 Nonparametric regression and quantile regression
60J60 Diffusion processes
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References:

[1] Baraud Y. (2000) Model selection for regression on a fixed design. Probability Theory and Related Fields 117: 467–493 · Zbl 0997.62027 · doi:10.1007/PL00008731
[2] Barron A., Birgé L., Massart P. (1999) Risk bounds for model selection via penalization. Probability Theory and Related Fields 113: 301–413 · Zbl 0946.62036 · doi:10.1007/s004400050210
[3] Birge L., Massart P. (2001) Gaussian model selection. Journal of the European Mathematical Society 3: 203–268 · Zbl 1037.62001 · doi:10.1007/s100970100031
[4] Chenstov N.N. (1962) On estimation of unknown density by observations. Reports of Academy of Sciences of USSR 147(1): 45–48
[5] Efroimovich S.Yu. (1999) Nonparametric curve estimation. Methods, theory and applications. Springer, Berlin
[6] Fourdrinier D., Pergamenshchikov S. (2007) Improved model selection method for a regression function with dependent noise. Annals of the Institute of Statistical Mathematics 59: 435–464 · Zbl 1122.62023 · doi:10.1007/s10463-006-0063-7
[7] Galtchouk, L., Pergamenshchikov, S. (2005). Efficient adaptive nonparametric estimation in heteroscedastic regression model. Preprint, IRMA, Université de Strasbourg Louis Pasteur, 020 ( http://www.univ-rouen.fr/LMRS/Persopage/Pergamenchtchikov ). · Zbl 1293.62091
[8] Galtchouk L., Pergamenshchikov S. (2006) Asymptotically efficient estimates for nonparametric regression models. Statistics and Probability Letters 76(8): 852–860 · Zbl 1089.62044 · doi:10.1016/j.spl.2005.10.017
[9] Gill R.D., Levit B.Y. (1995) Applications of the van Trees inequality: a Bayesian Cramér–Rao bound. Bernoulli 1: 59–79 · Zbl 0830.62035 · doi:10.2307/3318681
[10] Golfed S.M., Quandt R.E. (1972) Nonlinear methods in econometrics. North-Holland, Amsterdam
[11] Golubev G. (1982) Minimax filtering of functions. Problems of Transimission Information 4: 67–75 · Zbl 0514.62102
[12] Ibragimov I.A., Hasminskii R.Z. (1981) Statistical estimation: Asymptotic theory. Springer, New York · Zbl 0705.62039
[13] Kabanov Yu.M., Pergamenshchikov S.M. (2003) Two-scale stochastic systems. Asymptotic analysis and control. Springer, New York · Zbl 1033.60001
[14] Konev V.V., Pergamensgchikov S.M. (2003) Sequential estimation of the parameters in a trigonometric regression model with the gaussian coloured noise. Statistical Inference for Stochastic Processes 6: 215–235 · Zbl 1031.62062 · doi:10.1023/A:1025875212695
[15] Liptser R.Sh., Shiryaev A.N. (1977) Statistics of random processes. I. General theory. Springer, New York
[16] Nemirovskii A. (2000) Topics in non-parametric statistics. Lecture Notes in Mathematics 1738: 85–277
[17] Pinsker M.S. (1981) Optimal filtration of square integrable signals in gaussian white noise. Problems of Transimission Information 17: 120–133 · Zbl 0452.94003
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