One-step Consistent Estimators For The Partially Linear Models

English AbstractWe study the partially linear models,assume that the response variables de-pend on X= (x1,x2,...,xp)T and Z,and the relationship between Y and X is lin-ear relationship,the relationship between Y and Z is nonlinear relationship,so the par-tially linear model: Yi＝XiTβ＋g(Zi)＋εiwhere E(εi|Xi,Zi)=0,σi2=Var(εi|Xi,Zi),(i=1,...,n)X and Z are random variables.β= (β1,β2,....βp)T are unknown parametric vari-ables, g(·) is an unknown function. As part of the linear regression model contains both the parameter section, also includes some non-parametric. so it is more flex-ible than the linear model, we are interested in the linear part Xiβ,it reflects our interested in the relationship between variables, nonlinear part g(Zi) of the model increase in the adaptability of the model. This model is a generaliza-tion of the form, it contains many very important in Semi-parametric Models. Precisely because of this, many scholars began to be interested in this model, the parameters of this model and the non-parametric nature.Qian Weimin use two-stage method of estimation studied this model,and gave the consistency of the estimator. This method need the relationship between the. two pa-rameters a.β,assuming the parameterβis known, obtain the estimation of g(z,β) which de-pend onβ. Then using the original model.we can obtain the estimationβof param-eterβ. This method is cumbersome, and two-step estimation increase the error and the variance in the estimation, so I would like to find a way to re-duce the error caused by multiple estimation, I want to get the final estimation us-ing one-step estimation. This requires that we first form of non-linear function with parameters forms, the more common approach to the representation of a poly-nomial approximation, orthogonal. series approximation, etc. This article we use or-thogonal series approximation, this will reduce the number of iterations.Such rep-resentation may have some deviations from the real form of the function,a arti-cle of Yanyun Ma semi-parameters linear model,the improved weighted estima-tion equation can solve this problem. This article's main idea is that first ap-proximation the function g(z) by orthogonal series, then use the weighted estima-tion equation estimate the parameterβof the model.