On a Principal Varying Coefficient Model

Abstract

We propose a novel varying coefficient model (VCM), called principal varying coefficient model (PVCM), by characterizing the varying coefficients through linear combinations of a few principal functions. Compared with the conventional VCM, PVCM reduces the actual number of nonparametric functions and thus has better estimation efficiency. Compared with the semivarying coefficient model (SVCM), PVCM is more flexible but with the same estimation efficiency when the number of principal functions in PVCM and the number of varying coefficients in SVCM are the same. Model estimation and identification are investigated, and the better estimation efficiency is justified theoretically. Incorporating the estimation with the L ₁ penalty, variables in the linear combinations can be selected automatically, and hence, the estimation efficiency can be further improved. Numerical experiments suggest that the model together with the estimation method is useful even when the number of covariates is large. Supplementary materials for this article are available online.

Keywords:

• Local linear estimator
• L ₁ penalty
• Principal function
• Profile least-squares estimation
• Semivarying coefficient model

Acknowledgments

Wang’s work was supported in part by the National Natural Science Foundation of China (11131002, 11271032), Fox Ying Tong Education Foundation, and the Center for Statistical Science at Peking University. Xia’s work was partially supported by a grant from the National University of Singapore (R-155-000-121-112). Guohua Jiang’s work was supported in part by the National Natural Science Foundation of China (71132004). The authors are very grateful for the helpful comments of three referees, the associate editor, and the editor.