Local Polynomial Estimation On Semivarying Coefficient Models

In nonparametric regression,the methods of estimation include kernel,local polynomial and smoothing spline. These methods deal well with one-dimension dates. However,with the increase of dimension,the sample pointswe can get become very sparse,which results in the conventional methodsin nonparametric regression showing uselessness to high-dimension dates.Varying-coefficient model is designed to cope with difficulties in regressionof high-dimension dates. Varying-coefficient model not only inherits thetraits of robust partially from nonparametric models,but also keeps theadvantage of explicit in linear models. Therefore,the researches of varying-coefficient model have been given most attention recently and extensive ap-plications of varying-coefficient models into biometrics and medicine havebeen implemented successfully.The varying-coefficient model is defined by the following model: Y=sum from j=1 to pα_j(U)X_j+ε,Whereα_j(U),j=1,…,p are unknown univariate coefficient,εis randomerror. For given covariates (U, X₁,…, X_p)^T and response variable Y withE(ε|U, X₁,…, X_p)=0 and Var(ε|U, X₁,…, X_p)=σ²(U). In order to simplify the model,the investigators proposed the following model whenthey test if some coefficient functions are constants. Y=sum from j=1 to pα_j(U)X_j+sum from j=p+1 to p+mα_jX_j+ε,This model is called a semivarying coefficients model, which is an usefulextension of the varying coefficient model. In this paper, we discussedmainly the semivarying coefficients model.The main context are the following: In Chapter 1,we provide somenormal methods in nonparametric regression and dwell on the current re-searches of varying coefficient models and semivarying coefficient models.In Chapter 2, the estimators of constant coefficients are given by localpolynomial method and averaged method. A two-step method is takenfor the estimation of coefficient functions. In the first step,replacing theunknown constant coefficients,we obtain a varying coefficient model; In thesecond step, the estimator of coefficient functions is given by local polyno-mial method. The asymptotic normalities of these estimators are given. InChapter 3,The proposed methods are illustrated by a simulation,and themethods are shown ideal.