Separation of covariates into nonparametric and parametric parts in high-dimensional partially linear additive models

Determining which covariates enter the linear part of a partially linear additive model is always challenging. It is more serious when the number of covariates diverges with the sample size. In this paper, we propose a double penalization based procedure to distinguish covariates that enter the nonparametric and parametric parts and to identify insignificant covariates simultaneously for the "large p small n" setting. The procedure is shown to be consistent for model structure identification, it can identify zero, linear, and nonlinear components correctly. The resulting estimators of the linear coefficients are shown to be asymptotically normal. We discuss how to choose the penalty parameters and provide theoretical justification. We conduct extensive simulation experiments to evaluate the numerical performance of the proposed methods and analyze a gene data set for an illustration.