Estimation in partially linear models and numerical comparisons

Partially linear models with local kernel regression are popular non-parametric techniques. However, bandwidth selection in the models is a puzzling topic that has been addressed in literature with the use of undersmoothing and regular smoothing. In an attempt to address the strategy of bandwidth selection, we review profile-kernel based and backfitting methods for partially linear models, and justify why undersmoothing is necessary for backfitting method and why the "optimal" bandwidth works out for profile-kernel based method. We suggest a general computation strategy for estimating nonparametric functions. We also employ the penalized spline method for partially linear models and conduct intensive simulation experiments to explore the numerical performance of the penalized spline method, profile and backfitting methods. A real example is analyzed with the three methods.