The additive model with different smoothness for the components

Abstract

We consider an additive regression model consisting of two components $f^0$ and $g^0$, where the first component $f^0$ is in some sense "smoother" than the second $g^0$. Smoothness is here described in terms of a semi-norm on the class of regression functions. We use a penalized least squares estimator $(\hat{f},\hat{g})$ of $(f^0,g^0)$ and show that the rate of convergence for $\hat{f}$ is faster than the rate of convergence for $\hat{g}$. In fact, both rates are generally as fast as in the case where one of the two components is known. The theory is illustrated by a simulation study. Our proofs rely on recent results from empirical process theory.

26 pages, 4 figures

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