Functional aggregation for nonparametric regression

We consider the problem of estimating an unknown function f from N noisy ob servations on a random grid. In this paper we address the following aggrega tion problem: given M functions f(1),...,f(M) find an "aggregated" estimato r 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* wit h expected L-2 accuracy O(N(-1/)4 ln(1/4) M). We show that this approximati on rate cannot be significantly improved. We discuss two specific applications: nonparametric prediction for a dynami c system with output nonlinearity and reconstruction in the Jones-Barron cl ass.