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

摘要

We consider the standard additive regression model consisting of two components f(0) and g(0). The first component f(0) is assumed to be in some sense "smoother" than the second g(0). It is known that in that case one can construct estimators that estimate the smoother component f(0) with fast minimax rate as if the non-smooth component g(0) were known. Our contribution shows that this phenomenon also occurs when one uses the penalized least squares estimator ((f) over cap, (g) over cap) of (f(0), g(0)). 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. (C) 2015 Elsevier B.V. All rights reserved.