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Oracle Wiener filtering of a Gaussian signal
Teoriâ slučajnyh processov, Tome 17 (2011) no. 2, pp. 16-24 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the problem of filtering a Gaussian process whose trajectories, in some sense, have an unknown smoothness $\beta_0$ from the white noise of small intensity $\epsilon$. If we knew the parameter $\beta_0$, we would use the Wiener filter which has the meaning of oracle. Our goal is now to mimic the oracle, i.e., construct such a filter without the knowledge of the smoothness parameter $\beta_0$ that has the same quality (at least with respect to the convergence rate) as the oracle. It is known that in the pointwise minimax estimation, the adaptive minimax rate is worse by a log factor as compared to the nonadaptive one. By constructing a filter which mimics the oracle Wiener filter, we show that there is no loss of quality in terms of rate for the Bayesian counterpart of this problem - adaptive filtering problem.
Keywords: Bayesian oracle, Gaussian process, minimax pointwise risk, Wiener filter. 
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A. Babenko; E. Belitser. Oracle Wiener filtering of a Gaussian signal. Teoriâ slučajnyh processov, Tome 17 (2011) no. 2, pp. 16-24. http://geodesic.mathdoc.fr/item/THSP_2011_17_2_a1/

[1] A. Babenko, E. Belitser, “On posterior pointwise convergence rate of a Gaussian signal under a conjugate prior”, Statist. Prob. Lett., 79 (2009), 670–675

[2] Probl. Inf. Transm., 44, 321–332

[3] E. Belitser, B. Levit, “On the empirical Bayes approach to adaptive filtering”, Math. Meth. Statist., 12 (2003), 131–154

[4] I. A. Ibragimov, R. Z. Khasminski, Asymptotic Theory, Springer, New York rata, 1981

[5] I. Johnstone, Function Estimation and Gaussian Sequence Models Monograph draft, , 2009 http://www-stat.stanford.edu/ĩmj/

[6] O. V. Lepski, “One problem of adaptive estimation in Gaussian white noise”, Theory Probab. Appl., 35 (1990), 459–470

[7] O. V. Lepski, “Asymptotic minimax adaptive estimation. 1. Upper bounds”, Theory Probab. Appl., 36 (1991), 645–659

[8] O. V. Lepski, “Asymptotic minimax adaptive estimation. 2. Statistical model without optimal adaptation. Adaptive estimators”, Theory Probab. Appl., 37 (1992), 468–481

[9] X. Li and L. H. Zhao, “Bayesian nonparametric point estimation under a conjugate prior”, Statist. Prob. Lett., 58 (2002), 23–30

[10] H. Robbins, “An empirical Bayes approach to statistics”, Proc. 3rd Berkeley Symp. on Math. Statist. and Prob., v. 1, Univ. of California Press, Berkeley, 1956, 157–164

[11] A. Tsybakov, “Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes”, Ann. Statist., 26 (1998), 2420–2469

[12] A. Tsybakov, Introduction to nonparametric estimation, Springer, 2009