Local Polynomial Estimation Method On Adaptive Varying-coefficient Model

Varying-coefficient model is a kind of important statistical model, which was developed in the 1990s. It was first proposed by Hastie, Tibshirani(1993). In nonparametric regression, the methods of estimation include kernel, local polynomial, smoothing spine and the series. These methods deal well with one dimension data. However, the sample points we can get become very few with the increase of dimension, which leads to the conventional methods in nonparametric regression showing inefficiency to the high dimension data. Owing to a lot of the high dimension data is often happened, the analysis of high dimension data is the question people have been concerned about. Varying-coefficient model is designed to cope with difficulties in regression of high dimension data by Statisticians. Varying-coefficient model not only keeps the characteristics of robust partially from nonparametric models, but also keeps the traits of explicit in linear models.In this paper, we consider adaptive varying-coefficient model which is an important varying-coefficient model. Fan, J., Yao, Q.W. and Cai, Z.W.(2000)have proposed the adaptive varying-coefficient model and discussed its properties. In practice, the model has already been widely applied to areas such as medical science, biology, economics, finance.The adaptive varying-coefficient model is defined by the following model: is a random sample of incomplete data from this model, unknown quantityβ∈RP,Xi=(Xi1,Xi2,…,Xip)T, Xi0=1, are called the coefficient functions, are errors with independent identical distributions and E(εi)=0, Var(εi)=σ2.The estimation theory of varying-coefficient model has been quite mature, such as:Jianqing Fan and Zhang W.Y(1999) gives local polynomial two-step estimation method; Qingguo Tang, Jinde Wang (2005) put forward one-step estimation method to estimate unknown function with different degrees of smoothness, Qingguo Tang(2007)puts forward a new local linear estimation method to estimate unknown function which is not too smooth, but articles about adaptive varying-coefficient model estimation research is rarely, therefore, the research on estimation method of adaptive varying-coefficient model has certain theoretical value.Relative to the statistical inference of varying-coefficient model, in adaptive varying-coefficient model, we are not only need to estimate its coefficient functions, but also the direction of its projection direction. Therefore, the inference method of varying-coefficient model can not be directly applied to adaptive varying-coefficient model. In this paper, we discuss the coefficient functions estimation of adaptive varying-coefficient model, apply the local polynomial method into adaptive varying-coefficient model, and discuss the asymptotic properties of estimators. Then, we estimate the projection direction.The main content is as follows: In the first chapter, we mainly introduce adaptive varying-coefficient model and some non-parametric methods.In the second chapter, in a condition of givenβ, first of all, to coefficient functions with the same degree of smoothness, we give their estimation by using the local linear estimation method which is introduced in the first chapter, and prove the asymptotic normality. Secondly, we estimateβin the condition of given coefficient functions.In the third chapter, in a condition of givenβ, firstly, to the pth-coefficient functions with the high degree of smoothness, we use local polynomial estimation method to give its one-step and two step estimation, then, by calculating, we can get the asymptotic deviations and asymptotic variances of two estimates respectively, and draw the two-step estimation is superior to one-step estimation. Secondly, for other p-1 coefficient functions with lower degree of smoothness, we get their local linear estimations and two-step estimations by using local linear estimation method. We find that the asymptotic deviations are the same, but asymptotic variances of two-step estimation are less than the local linear estimation's, that is, the two-step estimation is superior to local linear estimation. Finally, we use one-step iterative method to estimateβin the condition of given coefficient functions.