Estimation and variable selection in partial linear single index models with error-prone linear covariates

Abstract

We study the estimation and variable selection for a partial linear single index model (PLSIM) when some linear covariates are not observed, but their ancillary variables are available. We use the semiparametric profile least-square based estimation procedure to estimate the parameters in the PLSIM after the calibrated error-prone covariates are obtained. Asymptotic normality for the estimators are established. We also employ the smoothly clipped absolute deviation (SCAD) penalty to select the relevant variables in the PLSIM. The resulting SCAD estimators are shown to be asymptotically normal and have the oracle property. Performance of our estimation procedure is illustrated through numerous simulations. The approach is further applied to a real data example.

Keywords:

• ancillary variables
• error-prone
• local linear smoothing
• profile least square method
• SCAD
• single-index

Acknowledgements

Zhang's research is supported by Natural Science Foundation of SZU (801, 00036112), China and NSFC grant 11101157 of China. Wang's research is supported by the NSFC grant 11101063 of China. Gai's research is supported by the NNSF grant 11201499 of China. The paper is partially supported by the NNSF grant 11201306, China, the Innovation Program of Shanghai Municipal Education Commission (13YZ065) and the Fundamental Research Project of Shanghai Normal University (SK201207). The authors greatly thank the Editor, an Associated Editor and two referees for their constructive comments that substantially improved an earlier version of this paper.