Penalized least squares estimation in the additive model with different smoothness for the components

We consider the standard additive regression model consisting of two components f0f0 and g0g0. The first component f0f0 is assumed to be in some sense “smoother” than the second g0g0. It is known that in that case one can construct estimators that estimate the smoother component f0f0 with fast minimax rate as if the non-smooth component g0g0 were known. Our contribution shows that this phenomenon also occurs when one uses the penalized least squares estimator (fˆ,gˆ) of (f0,g0)(f0,g0). We describe smoothness in terms of a semi-norm on the class of regression functions. This covers the case of Sobolev smoothness where the penalized estimator is a standard spline estimator. The theory is illustrated by a simulation study. Our proofs rely on recent results from empirical process theory.