Statistica Sinica 21 (2011), 1225-1248
doi:10.5705/ss.2009.140

ESTIMATION AND VARIABLE SELECTION FOR

SEMIPARAMETRIC ADDITIVE PARTIAL LINEAR MODELS

Xiang Liu$^1$, Li Wang$^2$ and Hua Liang$^1$

$^1$University of Rochester and $^2$University of Georgia

Abstract: Semiparametric additive partial linear models, containing both linear and nonlinear additive components, are more flexible than linear models, and they are more efficient compared to general nonparametric regression models because they reduce the ``curse of dimensionality". In this paper, we propose a new estimation approach for these models, in which we use polynomial splines to approximate the additive nonparametric components and derive the asymptotic normality for the resulting estimators of the parameters. We also develop a variable selection procedure to identify significant linear components using the smoothly clipped absolute deviation penalty (SCAD), and we show that the SCAD-based estimators of non-zero linear components have an oracle property. Simulations are performed to examine the performance of our approach as compared to several other variable selection methods, such as the Bayesian Information Criterion and Least Absolute Shrinkage and Selection Operator (LASSO). The proposed approach is also applied to data from a nutritional epidemiology study.

Key words and phrases: BIC, LASSO, penalized likelihood, regression spline, SCAD.