SCAD-penalized regression in high-dimensional partially linear models

TL;DR

This paper develops a method combining SCAD penalty and spline estimation for variable selection and estimation in high-dimensional partially linear models, achieving oracle properties and demonstrated through simulations and data analysis.

Contribution

It introduces a novel approach using SCAD penalty with spline estimation for simultaneous variable selection and estimation in high-dimensional partially linear models, with proven oracle properties.

Findings

Consistent variable selection and estimation for both linear and nonparametric parts.

SCAD-penalized estimators are asymptotically normal with oracle properties.

Method performs well in finite samples as shown by simulations and data analysis.

Abstract

We consider the problem of simultaneous variable selection and estimation in partially linear models with a divergent number of covariates in the linear part, under the assumption that the vector of regression coefficients is sparse. We apply the SCAD penalty to achieve sparsity in the linear part and use polynomial splines to estimate the nonparametric component. Under reasonable conditions, it is shown that consistency in terms of variable selection and estimation can be achieved simultaneously for the linear and nonparametric components. Furthermore, the SCAD-penalized estimators of the nonzero coefficients are shown to have the asymptotic oracle property, in the sense that it is asymptotically normal with the same means and covariances that they would have if the zero coefficients were known in advance. The finite sample behavior of the SCAD-penalized estimators is evaluated with simulation and illustrated with a data set.