## 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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### References:

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