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Functional aggregation for nonparametric regression. (English) Zbl 1105.62338
Summary: We consider the problem of estimating an unknown function \(f\) from \(N\) noisy observations on a random grid. In this paper we address the following aggregation problem: given \(M\) functions \(f_1,\dots, f_M\), find an “aggregated” estimator which approximates \(f\) nearly as well as the best convex combination \(f^*\) of \(f_1,\dots, f_M\). We propose algorithms which provide approximations of \(f^*\) with expected \(L_2\) accuracy \(O(N^{-1/4}\ln^{1/4} M)\). We show that this approximation rate cannot be significantly improved. We discuss two specific applications: nonparametric prediction for a dynamic system with output nonlinearity and reconstruction in the Jones-Barron class.

MSC:
62G08 Nonparametric regression and quantile regression
62L20 Stochastic approximation
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[21] LMC, 51 rue de Mathématiques Domaine Universitaire, BPS3 Grenoble, Cedex 9 France E-mail: juditsky@inrialpes.fr Faculty of Industrial Engineering and Management at Technion Technion, Israel Institute of Technology Technion City, Haifa 32000 Israel E-mail: nemirovs@i.e.technion.ac.il
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