Wavelet threshold estimation for additive regression models

Abstract

Additive regression models have turned out to be useful statistical tools in the analysis of high-dimensional data. The attraction of such models is that the additive component can be estimated with the same optimal convergence rate as a one-dimensional nonparametric regression. However, this optimal property holds only when all the additive components have the same degree of "homogeneous" smoothness. In this paper, we propose a two-step wavelet thresholding estimation process in which the estimator is adaptive to different degrees of smoothness in different components and also adaptive to the "inhomogeneous" smoothness described by the Besov space. The estimator of an additive component constructed by the proposed procedure is shown to attain the one-dimensional optimal convergence rate even when the components have different degrees of "inhomogeneous" smoothness.