The additive model with different smoothness for the components

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$(f^0, g^0)$andshowthattherateofconvergencefor$\hat f $ is faster than the rate of convergence for $\widehat{g}$. 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.