Estimation In Functional-Coefficient Partially Linear Regression Model With Data Missing At Random

A functional-coefficient partially linear regression model is considered when the responses are missing at random(MAR).For exploring the influence of the input on the output of a system and exploiting the internal interaction of the outputs measured at different time, furthermore forecasting the output of the system, the functional-coefficient partially linear regression model is proposed by Zhang Riquan. It can be seen as a functional-coefficient regression model with different smoothing variables. It is also a generalization model of the partially linear regression model by replacing the parameters in the partially linear regression model with some functions of the covariates. It includes two parts: the constant part function, which shows the influence of the input on the output, and the varying coefficient part, the coefficient functions of which show the internal interaction of the outputs measured at different time. In reality, some parts of the explored matrix is often not responsive. Consider the functional-coefficient partially linear regression model in this paper{(Y_i,δ_i,X_i,U_i,Z_i)}_i=1ⁿ is a random sample of incomplete data from the model, where for each i,δ_i=1 if the Y_i is observed andδ_i=0 otherwise, and Z_i=(Z_il,…Z_ip)^T with T denoting the transpose of a matrix or a vector;{α_j(·)}_j=0^p are called the coefficient functions;{ε_i}_i=1ⁿ are errors with independent identical distributions and E(ε_i)=0,Var(ε₁)=σ².Throughout this paper, we assume that Y is missing at random (MAR). The MAR assumption implies thatδand Y are conditionally independent given X,U,Z,that is,P(δ=1|Y,X,U,Z)=P(δ=1|X,U,Z)≡π(X,U,Z)>0.Furthermore, in the context of the response variable missing at random, the following three problems are discussed.1) The local estimation with the complete-case data. The local linear method and averaged technique are employed to give the estimators of all functions. The asymptotical normalities of all estimators are given.2) The locally weighed linear estimation. The local linear method and averaged technique are also employed to give the estimators of all functions. The asymptotical normalities of all estimators are given.3) The local regression estimation with the imputed vales. The imputation method consists of two steps. The first step involves imputing missing response values. In the second step, we substitute Y_i byThen, the same estimation techniques based on imputed values are applied to obtain the efficient estimators of all functions. The asymptotical normalities of all estimators are given.