Estimation On Varying-coefficient Partially Linear Models With Missing Data

For the expensive experimental, the object of study is unable to continue investigation and so on. We often have the problem of missing data. In this paper, we consider the estimation on varying coefficient partially linear model with missing data. Model is considered as follows: Y=α(X)＋(?)αi(U)Zi＋εwhere the Y is a real response variable which is missing at random; X∈R、U∈R andZ=(Z1,…, Zp)T∈Rp are the covariate variable; ε is error with independent identical distributions and E(ε)=0, Var(ε)=σ2; the constant function α (·) and αi(·)(i=1,2,…p) are unknown measurable functions. Defined δj is an indicator variable which is used to indicate whether Yj has an observation, δj=1if Yj is observed and δj=0if Yj is not observed.In this paper, we discuss the estimation problem of the functions α(X) and αi(U)(i=1,2,…p). First, remove the incomplete date,we use the inverse probability as the weighted to weight the complete data;. We can estimate with the weighted data, using the local linear method and averaged method to give the estimators. The second method is to supplement the missing dates by using the average of the observations, using the local linear method and averaged method to give the estimator, and proving the asymptotic property..Main content of this article is as follows:In the first chapter, we mainly introduce the background, basic methods, idea and give a brief introduction to the basics.In the second chapter, we mainly use a weighted method to deal with missing data, weighted the complete data; remove the incomplete date. We can estimate with the weighted data, using the local linear method and averaged method to have the estimators. The consistency of all estimators is gained.In the third chapter, we consider the mean imputation method. First, using the average supplement missing values, and we use the local linear method and averaged method to give the estimators of the constant function and coefficient functions. The asymptotic normality of each estimator is given. This method overcome the shortcomings of the second chapter which only use part of the data, making full use of the data.We can see that this mean imputation method has a better estimator of the effect.