The additive model with different smoothness for the components

TL;DR

This paper introduces an additive regression model with components of different smoothness levels, demonstrating that the estimator for the smoother component converges faster, supported by empirical process theory and simulations.

Contribution

It develops a penalized least squares estimator for additive models with components of varying smoothness, showing improved convergence rates for the smoother component.

Findings

Faster convergence rate for the smoother component

Both components achieve near-optimal convergence rates

Empirical process theory underpins the theoretical results

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.