Variable Bandwidth And Local M-estimators Of Varying-coefficients Models

The article consider a varying-coefficient model:Which Y is a response variable,X=(X₁,…,X_p)^T are random variables,a_j(T)(j=1,…,p)are the unknown functions with the same smooth degree,(?) is random error satisfying E((?)|T,X)=0,Var((?)|T,X)=σ²(T).There are several methodsfor the function coefficient estimates in model(1),When a_j(·)(j=1,…,p)havethe same smooth degrees,literature[1]gives a partial least squares method is a simple and useful way,which get the optimal estimator.Hastie(1993)proposedthe smooth spline and the kernel method in article[2];Fan(2000),Tsang(2001)proposed the local polynomial method and the two step smooth spline method.Qingguo Tang(2005)Proposed one step estimate method used to estimate unknown functions which have different smoothness of variable coefficient model.Yiqiang Lu(2003)estimate a_j(·)'s in model(1)through B spline,discuss the B spline M-estimateconvergence rate under the repeat observation situation.Local polynomial regression methods have been demonstrated as effective nonparametric smoothers.They have advantages over popular kernel methods,in termsof the ability of design adaptation and high asymptotic efficiency.Moreover,thelocal polynomial regression smoothers can adapt to almost all regression settings and cope very well with the edge effects.A drawback of these local regression estimators is,however,lack of robustness,and M-type of regression estimators are natural candidates for achieving desirable robustness properties.In this paper,thevariable bandwidth and one step local M approach is employed to estimate the coefficient functions in varying coefficient models,The proposed method inherits theadvantages of local polynomial regression and overcomes the shortcoming of lack of robustness of least-squares techniques.The use of variable bandwidth enhances the flexibility of resulting local M-estimators and makes them possible to cope wellwith spatially inhomogeneous curves,heteroscedastic errors and nonuniform design densities.Under appropriate regularity conditions,we get the consistency and asymptotically normal of estimators of coefficient functions.The local M-regressioninherits many nice statistical properties from the local least-squares regression,unlike the local least-squares regression,the local M-regression estimators are definedimplicitly and numerical implementation requires an iterative scheme.This may add large computational burden on the procedure and makes it less attractive.To reduce the computation,This paper discuss one-step local M-regression estimator.This estimator shares the same computational expediency as the local least-squares estimator,and possesses the same asymptotic performance as the local M-regressionestimator when the initial estimator well behaves.In other words,the local onestepM-regression estimator,while robustifying the local least-squares estimator,truly inherits all good properties from the local least-squares estimator,in terms of not only asymptotic performance but also computational expediency.At last,simulation study of our estimations by using Matlab are showed in this paper.According to the results,we conclude that our methods are good.Theone-step and two-step estimators improve largely over the least-squares method.Forreasonably large bandwidths,the one-step and two-step local M-regression estimatorsare nearly as efficient as the fully iterative method.