A Note on the Optimal Smoothing in Partial Linear\centerline Models with Penalized Least Squares Estimator

Partially linear models are assumed to be linearly related to one or more variable, but the relation to an additional variable or variables is not assumed to be easily parameterized. One primary approach to estimate the parameter and nonparametric part is the method of penalized least squares method, generalized cross-validation (GCV) approach is a popular method for selecting the smoothing parameters. However, the optimality of GCV in the partial linear model with penalized least squares has not been proved. In this article, we provide the support for using GCV through its optimality of the smoothing parameter. Simulation studies are employed to investigate the empirical performance of generalized cross-validation and that of cross-validation for comparison in the context.