Parametric component detection and variable selection in varying-coefficient partially linear models

In this paper we are concerned with detecting the true structure of a varying-coefficient partially linear model. The first issue is to identify whether a coefficient is parametric. The second issue is to select significant covariates in both nonparametric and parametric portions. In order to simultaneously address both issues, we propose to combine local linear smoothing and the adaptive LASSO and penalize both the coefficient functions and their derivatives using an adaptive L1L1 penalty. We give conditions under which this new adaptive LASSO consistently identifies the significant variables and parametric components along with estimation sparsity. Simulated and real data analysis demonstrate the proposed methodology.