Estimation Problems Of Functional-Coefficient And Partial Linear Models

At the recent decades the theories, methods and applications of the partial linear models and the functional-coefficient partial linear models, which have been focused on, have been developed rapidly due to their abundant research contents and wide applications. In the paper the functional-coefficient partial linear models and the proportional functional-coefficient linear models are discussed, and the asymptotical properties of the estimates of the parameters and the unknown functions in these models are established. Also, the robust M-estimate for the partial linear models is established. The following is a brief introduction of each chapter.In the first chapter we give a brief description of the estimating method of univariate non-parametric regressing models. It includes three types: the local modelling approach, the orthogonal series approach and the spline approach. In the paper the local polynomial method which is contained the local modelling approach is employed to estimate the unknown functions. Also, we make a discuss on multivariate regressing models. Since there exists a serious problem, which is about dimension. When the dimension increases, the speed of convergence slows and the estimator is not steady. In statistics this is called curse of dimensionality. In order to solve this problem, all kinds of models are proposed to reduce the dimension. We propose two functional- coefficient models: the functional-coefficient partial linear models and the proportional functional-coefficient linear models. We make a retrospection of the robust M-estimate and continue to discuss the robust M-estimate used in local linear model. The hypothesis of the independence of sample is convenient to make research, but in some cases it is not reasonable. While dealing with economical data, the sample observed is dependent on each other. We sum up the types of the extant mixing dependent, a-mixing dependent is weaker than other mixing dependent. Many stochastic processes which include many time series can satisfy the conditions of a-mixing dependent. In the paper we make a study of the data which is independent and also the data which is a-mixing dependent.In the second chapter, the functional-coefficient partial linear model is discussed.Hastie and Tibshirani (1993) proposed the functional-coefficient model.where are some unknown functions; Y is response variable; X Rq arid Z e Rp are covariables; e is a random error with E(e) = 0.The model is a general model. But it has no practicable meaning when q > 1. To solve this problem, statistician gave out-many practicable models. When take q = 1, etc X is one dimensional variable, and let the constant part function P0() and coefficient functions have the same arguments, the model will be the one that we often use. In order to solve this problem, we propose the following models in which the constant part function and coefficient functions have different arguments:where Y e R is response variable, are covariebles. are some unknown measurable function from R to R. Weassume that Z and U are independent of X, and E = 0 in order to identifythe models. The models are called the functional-coefficient partial linear models.The functional-coefficient partial linear model is a general model. When for all j = is a constant, it becomes the partial linear model. When 0, it becomes functional-coefficient regression model proposed by Cai, Fan and Yao(2000). When for all j = 1, ???,p, it becomes nonparametric model.In the functional-coefficient partial linear model g(-) and have different argument, which makes us to estimate g(-) and even harder. In this chapter,we make deeper studies of the functional-coefficient partial linear models. We give out the estimator of the constant part function g(-) by local linear method. Using the estimator of the constant part function , we employ the local linear method to estimate coefficient function The method is called two-stage estimating method. The weak consistence, the uniform strong consistence and t...