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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 ,,f M , find an “aggregated” estimator which approximates f nearly as well as the best convex combination f * of f 1 ,,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:
62G08Nonparametric regression
62L20Stochastic approximation