Local Polynomial Eestimatiors In Semiparametric Regression Model

Semiparametric Regression Models, which are an integration of parametric and nonparametric regression models and therefore more practically useful, have been explored intensively in recent years. One important goal of the research is to obtain estimations of beta and g from Semiparametric Regression Models and to investigate their asymptotic behavior with given information. Lots of approaches ,like least squares, kernel smoothing, neighborhood priori, and wavelets estimation etc, have been proposed to solve this problem.This paper introduces two methods (Method 1: use nonparametric estimation first and then parametric estimation, modify nonparametric estimators finally. Method 2: use parametric estimation first and then nonparametric estimation, modify parametric estimators finally.) to estimate semiparametric regression models. Both methods adopt kernel smoothing for the nonparametric estimation part. In the paper, the estimators obtained from the above two methods are compared. However, kernel smoothing has an undesirable property, which is called edge effects. To deal with this deficiency, the paper considers using a newly proposed method, so-called local polynomial estimation, which avoids edge effects while keeps other merits of kernel smoothers. Local polynomial estimation is widely used for its other advantages like the adaptation to either fixed or random designs and other various design densities and it is the best linear estimator in the sense of minimax efficiency[seeFan(1993)] .Adopting local polynomial estimation, this paper makes some improvements for the second method of estimation for semiparametric regression models. The resulting estimators show not only good asymptotic properties but also boundary carpentry[Fan&Gijbels(1992)] that is remarkably different from the edge effects in the kernel smoothing. These advantages are further demonstrated by the numerical simulation in the paper.