Ultrahigh dimensional single index model estimation via refitted cross-validation

Abstract

This article develops a new estimation procedure for ultrahigh dimensional sparse single index models. We first use B-spline to approximate the link function, and a naive two-stage estimator can be applied for the estimation of single index models. However, the direct method may not perform well in ultrahigh dimensional data due to the spurious correlations, and the asymptotic results show that it may significantly underestimate the error variance and have greater estimation bias of the regressors coefficients. We further propose an accurate estimate for error variance and parameters in ultrahigh dimensional sparse single index model by effectively integrating sure independence screening and refitted cross-validation (RCV) techniques. The consistency and asymptotic properties of the resulting estimate are established under some regularity conditions. The simulation studies are carried out to study the finite sample performance of the newly proposed methods.

Keywords:

• Ultrahigh dimensional data
• sparse single index model
• spurious correlation
• refitted cross-validation

Acknowledgments

The content is solely the responsibility of the authors and does not necessarily represent the official views of NPSSFC. The authors thank the Editor, the Associate Editor, and the reviewers for their careful reading and constructive comments which have helped us to significantly improve the paper.

Additional information

Funding

This work was supported by the Key Program of National Philosophy and Social Science Foundation of China (NPSSFC) [21AJL010].