On minimax identification of nonparametric autoregressive models

We consider the problem of nonparametric identification for a multi-dimensional functional autoregression y(t) = f (y(t-1), ..., Yt-d) + e(t) On the basis of N observations of y(t). In the case when the unknown nonlinear function f belongs to the Barren class, we propose an estimation algorithm which provides approximations of f with expected L-2 accuracy O(N-1/4 In-1/4 N). We also show that this approximation rate cannot be significantly improved. The proposed algorithms are "computationally efficient" - the total number of elementary computations necessary to complete the estimate grows polynomially with N.