  In this paper we discuss the estimation of a nonparametric component $f_1$ of
a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + \epsilon$. We
allow the number $q$ of additive components to grow to infinity and we make
sparsity assumptions about the number of nonzero additive components. We
compare this estimation problem with that of estimating $f_1$ in the oracle
model $Z= f_1(X_1) + \epsilon$, for which the additive components
$f_2,\dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing
estimator of $f_1$ and state finite-sample bounds for the difference between
our estimator and some smoothing estimators $\hat f_1^{\text{(oracle)}}$ in the
oracle model. In an asymptotic setting these bounds can be used to show
asymptotic equivalence of our estimator and the oracle estimators; the paper
thus shows that, asymptotically, under strong enough sparsity conditions,
knowledge of $f_2,\dots,f_q$ has no effect on estimation accuracy. Our first
step is to estimate $f_1$ with an undersmoothed estimator based on
near-orthogonal projections with a group Lasso bias correction. We then
construct pseudo responses $\hat Y$ by evaluating a debiased modification of
our undersmoothed estimator of $f_1$ at the design points. In the second step
the smoothing method of the oracle estimator $\hat f_1^{\text{(oracle)}}$ is
applied to a nonparametric regression problem with responses $\hat Y$ and
covariates $X_1$. Our mathematical exposition centers primarily on establishing
properties of the presmoothing estimator. We present simulation results
demonstrating close-to-oracle performance of our estimator in practical
applications.
