Studies And Analysis On The Varying Coefficient Model

In the nonparametric regression the regression function is supposed to be from some function family, such as the smoothing functions. So the nonparametric regression needs few hypotheses and is very robust. As one of the leading branches of the modern statistics, the riouparametric regression analysis is widely applied to explore the relationship between the the response variable y and the covariable A'. All kinds of the methods are proposed to estimate the nonparametric regression function, such as the kernel method, the local polynomial estimators, the smoothing spline, the series estimators (B-spline estimator. Fourier series estimator, Wavelet series estimator). The essence of the above estimating methods is local estimator or local smoothing technique. In general, the non-parametric regression function is.well estimated by the above methods when the covariable X is one dimension. But the multivariable nonparametric regression function could not be well estimated by the local estimator because there is only a little data in the local fields of the high dimension regression variable X. This phenomenon is said to be 'the curse of dimension'. Due to a lot of the high dimension data is often happened, the analysis of high dimension data is one of the aspects in which a lot of statisticians are interested. Recently a lot of methods for high dimension data are discovered. There are essentially two approaches: the first is largely concerned with function approximation and the second with dimension reduction. The Example of the former are the additive model (Hastie andTibshirani, 1986), Partly linear model (Engle, et al ;1986) Projection Pursuit Regression(Friedman and Stuetzle, 1981). Example for the latter are dimension reduction, sliced inverse regression (Li, 1991)), graphical regression, Cook, 1994), Principal Hessian direction (Cook, 1998), minimum average variance estimation method (Xia, Y. et al. , 2002). The varying coefficients model, which is the function approximation method for high dimension, is discussed in this paper.A varying coefficients model has the general formwhere are regression variables, y is the response variable, e is random error, p are the unspecifiedsmoothing functions ,change the coefficient of the though the unspecified function . The dependence of on implies a special kind of interaction between each and . The variables s might be equal or not, even might be some Especially, if are same and denoted as , then model (1) is turned to beConveniently the B-spline estimators of function coefficients are described in the example of the model (2). The methods of B-spline estimators can be directly generalized to the model (1) by according to the characteristic of the B-spline estimators.Although more hypotheses are needed for the varying coefficient model than the mul-tivariable nonparametric regression, the varying coefficients model is rather than general. A lot of models appearing in the literatures, such as the additive model, the partly linear model, the linear model and so on, are considered as the particular examples of the varying coefficients model.The varying coefficients model is widely applied to the analysis of longitudinal data, nonlinear time series and the biologic data. Recently many of statisticians have been interested in the varying coefficients model because of its simplifying structure ,meaningful interpretation and wide application(Wu et al. , 1998;Fan and Zhang, 2000; Chiang, Rice and Wu.2001; Cai, Fan and Yao,2000 and so on).In this paper the function coefficients in the varying coefficients model are estimated by B-spline function. The rest of this paper is organized as follow.Chapter 1 is an exordium. This chapter mainly includes the smoothing meth-ods,smoothing parameter,the selection of smoothing parameter,the curse of dimension, B-spline function and the main content of this paper.Chapter 2 discusses the least square B-spline estimators of the function coefficients and their asymptotic character when the observational dat...