Semiparametric efficient estimation in high‐dimensional partial linear regression models

Abstract

We introduce a novel semiparametric efficient estimation procedure for high‐dimensional partial linear regression models to overcome the challenge of efficiency loss of the traditional least‐squares based estimation procedure under unknown error distributions, while enjoying several appealing theoretical properties. The new estimation procedure provides a sparse estimator for the parametric component and achieves the semiparametric efficiency as the oracle maximum likelihood estimator as if the error distribution was known. By employing the penalized estimation and the semiparametric efficiency theory for ultra‐high‐dimensional partial linear model, the procedure enjoys the oracle variable selection property and offers efficiency gain for non‐Gaussian random errors, while maintaining the same efficiency as the least squares‐based estimator for Gaussian random errors. Extensive simulation studies and an empirical application are conducted to demonstrate the effectiveness of the proposed procedure.