Efficient estimation and computation of parameters and nonparametric functions in generalized semi/non-parametric regression models

The efficiency of estimation for the parameters in semiparametric models has been widely studied in the literature (Bickel et al., 1993). In this paper, we study efficient estimators for both parameters and nonparametric functions in a class of generalized semi/non-parametric regression models, which cover commonly used semiparametric models such as partially linear models, partially linear single index models, and two-sample semiparametric models. We propose a maximum likelihood principle combined with the local linear technique for estimating the parameters and nonparametric functions. The proposed estimators of the parameters and a linear functional of the nonparametric functions are consistent and asymptotically normal and are further shown to be semiparametrically efficient. An efficient computational algorithm to achieve the maximization is proposed. Extensive simulation experiments show the superiority of the proposed methods. Three real data examples are analyzed and presented as an illustration.