Nonparametric estimation of composite functions.
Anatoli Juditsky, Oleg Lepski and Alexandre Tsybakov
Annals of Statistics 2007.

Abstract

We study the problem of nonparametric estimation of a multivariate function $g:\bR^d\to\bR$ that can be represented as a composition of two unknown smooth functions $f:\bR\to\bR$ and $G:\bR^d\to\bR$. We suppose that $f$ and $G$ belong to some known smoothness classes of functions and we construct an estimator of $g$ which is optimal in a minimax sense for the sup-norm loss. The proposed methods are based on aggregation of linear estimators associated to appropriate local structures, and the resulting procedures are nonlinear with respect to observations.