The Asymptotic Properties Of A Semiparametric Regression Model Under Fixed Design

A semiparametric regression model has been an important statistics model since 1980s. This kind of model includes not only a parametric component but also a nonparametric component. So it has the advantages of the parametric regression model and the nonparametric regression model. It has the more implements and stronger explanations than the pure parametric or nonparametric regression model .This paper considers an important semiparametric regression modelwhere is an unknown function, is an unkown parameter to be estimated. ei are iid. random errors with Bei = 0 and Eei2 = 2 < .In many practical problems, Xi usually are some nonrandom design points, that is , fixed design points. So, the purpose of this paper concentrates on the semiparametric regression model's large sample property-consitency when xi are the fixed design points.Unlike the normal two stages estimate method (the usual nonparametric weighted method combined with the least square estimate) , considering the characteristics of this model, this paper uses the least square estimate combining with the usual nonparametric weighted method and defines the estimators and n2 for the unknown parameter ,the unkown fuction g( ) and the unknown variance of errors 2 .The basic idea of the estimate method is ,firstly ,based on the linear model yi = x'i + ei,defining the least square estimator n of the linear model for the unkown parametric ;secondly, using the estimator n we 've got to substitute for in the original semiparametric regression model yi = x'i + g(xi) + ei and using the usual nonparametric weighted function method to define the estimator gn(-) for the unknown function g( ); finally ,defining the estimator 2 for the unknown variance of errors 2.For simplicity ,this paper focuses on the consistency of the model in one dimension case,that is d = 1. This paper also points out the consistency that can be generalized more than one dimension. So ,we achieve the large sample property -consistency of this class of model on the fixed design.In this paper,for fixed design points xi; under the assumption that the unknown function g is continuous function and the moment of random error exists and is finity,we discuss and show that the estimators n,gn and n2 for ,g and 2 have strong consistency, p th-mean consistency for more general nonparametric weighted fuction.