Penalty, post pretest and shrinkage strategies in a partially linear model

We addressed the problem of estimating regression coefficients for partially linear models, where the nonparametric component is approximated using smoothing splines and subspace information is available. We proposed pretest and shrinkage estimation strategies using the profile likelihood estimator as the benchmark. We examined the asymptotic distributional bias and risk of the proposed estimators, and assessed their relative performance with respect to the unrestricted profile likelihood estimator under varying degrees of uncertainty in the subspace information. The shrinkage-based estimators uniformly dominated the unrestricted profile likelihood estimator. The positive-part shrinkage estimator was shown to be more efficient than the others, and was robust against uncertain subspace information. We also compared the performance of penalty estimators with those of the proposed estimators via a Monte Carlo simulation, and found that the proposed estimators were more efficient. The proposed estimation strategies were applied to a real dataset to evaluate their practical usefulness. The results were consistent with those from theory and simulation.