Kernel Smoothing Estimation For Varying-coefficients EV Models

Formally the errors-in-variables (EV) models , also called measurement error (ME) models, are just the regression models with both dependent and independent variables being subjects to error . The study of EV models has a long history . Scientists encountered this model long before the twentieth century (Adcock , 1877 , 1878;Kummel, 1879) . Many works are given by the monograph of Fuller (1987) , which concems mainly the linear case . But because the special structure of EV models , the study is more difficult and the problem of existence of consistent estimate of parameters in EV model is more complicated than in the classical regression model (Cheng & Van Nass , 1999 ) . So far, the usual approach of the study on EV models are this : Firstly , some assumptions are made , for example , the model assumes a normal error and the variance of the measurement errors content some assumptions (for example , covariance matrix is positive definite matrix etc . ) There exists a huge literature under such assumptions in the model. The study of non-normal case are made by some scholars too. Another approach, aiming at avoiding such some what artificial assumptions is to take replicated observations and based on these observations , to establish estimates with good asymptotic properties .Varying-coefficient models (VCM) was propsed by Clevel-and, Grosse and Shyu (1992) , then discussed by Hastie and Tibshirani(1993) in detail . Up to now , it has made great influence in our world . In theory , the researchs of VCM has been given most attention recently;Furthermore , extensive applications of VCM into biometrics and medicine has implemented successfully .Considering there extist some cannot slighting errors in the observation of independent variable in practice ( for example errors come from measure tools and so on ) , we added EV models to varying-coefficient models ,which may express our problems more externally and the profounding meanings will be made in prcatice and theory. Then we can find a new research task : Varying-coefficient EV models. In this paper, we consider a varying-coefficient EV model which has following form:There into:which {Xi,yt) are random variables in Rp+X x Rl , fa,/,) cannot be observated accuratedly, and the observational results are (Xh Yi). Pj(t){j = 0,1, ■■■ ,p) are unknown limitary continuous functions , and fij(t) ± 0(J = 0,1,??■ ,p) , j8(0 called varying-coefficient . / are real-valued random variables , and its support is limitary close set, suppose it is [0,1] . (e,-, uJ)T are P + 2 dimensional random U.d errors vectors with Efa, uJ)t = 0, Cov(sh uJ)T = o2Ip+2, a2 > 0 is unknown . xt and ?,- , yi and e,- , xt and Bj are uncorrelation, all observation are independent.In this paper , The estimation of coefficientfunctions and variance of the measurement errors are constructed on this model by using kernel smoothing and generalized least square method . The main ideal are this : Firstly , we assume that coefficient functions take their mathematical expectations, which can be changed the model into normal linear model. We can get the one-step estimation of coefficient by using least square method. At this time, we dinned the one-step kernel estimation of coefficient functions by using p estimation resits . Then we change the model using the result we have got. The second-step estimation of coefficient we can get by using generalized least square . Now we defined the second-step kernel estimation of coefficient functions by using these results . The estimationof variance of the measurement errors are defined by the forehead results . Under some baisc regularity conditions, we get the strong consistency and uniform strong consistency of estimators of coefficient functions and the strong consistency of estimators of variance of the measurement error.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 .