SCAD-Penalized Regression in High-Dimensional Partially Linear Models

Summary. 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 be asymptotically normal with the same means and covariances that they would have if the zero coefficients were known in advance. Simulation studies are conducted to evaluate the finite sample behavior of the SCAD-penalized estimators. Key Words and phrases. Asymptotic normality, high-dimensional data, oracle property, penalized estimation, semiparametric models, variable selection,. Short title. High-dimensional PLM AMS 2000 subject classification. Primary 62J05, 62G08; secondary 62E2