Partial linear single-index models with additive distortion measurement errors

ABSTRACT

We study partial linear single-index models (PLSiMs) when the response and the covariates in the parametric part are measured with additive distortion measurement errors. These distortions are modeled by unknown functions of a commonly observable confounding variable. We use the semiparametric profile least-squares method to estimate the parameters in the PLSiMs based on the residuals obtained from the distorted variables and confounding variable. We also employ the smoothly clipped absolute deviation penalty (SCAD) to select the relevant variables in the PLSiMs. We show that the resulting SCAD estimators are consistent and possess the oracle property. For the non parametric link function, we construct the simultaneous confidence bands and obtain the asymptotic distribution of the maximum absolute deviation between the estimated link function and the true link function. A simulation study is conducted to evaluate the performance of the proposed methods and a real dataset is analyzed for illustration.

KEYWORDS:

• Confounding variable
• errors-in-variables
• SCAD
• single index

MATHEMATICS SUBJECT CLASSIFICATION:

• 62-02
• 62G08
• 62G20

Acknowledgments

The authors thank the editor, the associate editor, and a referee for their constructive suggestions that helped them to improve the early manuscript. Zhang Jun ’s research is supported by the National Natural Sciences Foundation of China (NSFC) [grant number 11401391] and the Project of Department of Education of Guangdong Province of China [grant number 2014KTSCX112]. Feng Zhenghui's research is supported by the Fundamental Research Funds for the Central Universities, China [grant number 20720171025].