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