Some Large-sample Properties Of Local Least-square Estimators In A Class Of Varying-coeffcient Models

Varying-coeffcient models(VCM) was proposed by Cleveland, Grosse and Shyu(1992), then discussed by Hastie and Tibshirani(1993) in detail. As we have known, estimation in nonparametric regression including kernel, local polynomial and smoothing spline deal well with one-dimension data. However, with the increase of dimension, the observational points we can get become very sparse, which results in the conventional methods in nonparametric regression showing uselessness to high-dimension data. We call this ph-nomenon "curse of dimension". VCM is designed to cope with the difficulties in regression of high-dimension data. On one hand, VCM not only inherits the traits of robust partially from nonparametric models, but also keeps the advantage of explicity in linear models; And on the other, VCM can be regarded as a universal model which can be turned into a variety of specified models. Therefore, the reserches of VCM have been given most attention recently; Furthermore, extensive applications of VCM into biometrics and medicine have implemented successfully.In this paper, we consider a class of VCM which has following form:where (xi,ti) are observational data, and xip = 1, (1 ≤ i ≤ n), βj(ti)(j ≥ 1) are unknown smoothing functions, and εi are independently identified distributional(i.i.d) random variables. Fan and Zhang(2000) obtained two-step estimator under the norms of local least-square. What the different method in this paper is that we firstly keep the pth function-coeffcient remained in order to reserve the information of model maximally, in other words, we regard it as known. Then we give the expansion of first p - 1 function-coeffcients and aquire the estimators of them. Finally, we return to the original model and get the pth estimator of function-cofficient. At the same time, the estimator of variance σ2 is obtained, and we disscussed some large-sample properties of above estimators.The whole paper is arranged as follows:Chapter 1 is the introduction. Section 1.1 introduces the backgrounds of development and provides some normal methods in nonparametric regression models. Section 1.2 and section 1.3 dwell on the current reserches in VCM and methods of estimation in this paper respectively. Chapter 2 gives the basic assumptions and lemmas which play a elementary role in proof of this paper.Chapter 3 obtained different asymptotic normal distributin of...