Some Results Of Two Kinds Of Covariate Measurement Errors Models

Variables in statistical models are often unable to be observed directly or accu-rately due to sampling errors,experimental errors and so on.Measurement error data are often encountered in economics,finance,biomedicine.If the least square method and maximum likelihood method are used to analyze the data with measurement error,the results are not unbiased and consistent.In order to make a reasonable statistical inference on this kind of data,statisticians use surrogate variables to establish mea-surement error models.The research on measurement error model is one of the most important issues.Relevant studies about this issue have been done by Brown(1982),Fuller(1987),Carroll(1995),Cheng and Ness(1992),He and Liang(2000),Cui(2004,2005,2006,2007)and so on.In this thesis,we are mainly concerned about covariate adjusted regression model and linear EV(errors-in-variables)model.The main content contains three parts.In the first part,the authors study the covariate adjusted linear regression model for situa-tions where predictors and response are observed after being distorted by multiplicative factor.The robust estimation of the model parameters is obtained.The model param-eters are estimated in two steps:in the first step,the basic idea is to transform the CAR model into a varying-coefficient regression model.The authors present the local linear L1-estimation method when the underlying error distribution deviates from a normal distribution.In the second step,the estimates of the regression coefficients can be obtained by targeting weighted averages of the smooth varying coefficient functions.We derive the consistency and asymptotic normality of the parameter estimates.Em-pirical likelihood ratio method based on L1 estimator is proposed.Based on bootstrap method,the hypothesis testing procedure is proposed.Simulations and empirical study are carried out.In the second part,the authors study the covariate-adjusted partially linear regression model.The response variables and prediction variables are regressed with respect to distorting covariate,and the kernel smoothing method is used to es-timate distorting functions,and the estimation of unobservable variables is obtained.It is proved that the empirical likelyhood ratio statistic is asymptotically chi-squared.The confidence intervals of parameters are constructed.Finally,the authors consider the robust statistical inference of linear EV model using instrumental variables.The linear regression of predictive variables and instrumental variables are extended to par?tial linear regression.The model parameters are estimated by the composite quantile regression method.The asymptotic normality of parameter estimators is proved.Due to the complexity of the asymptotic variance structure of estimators,a corrected empir-ical likelihood inference method is proposed.It is proved that the corrected empirical likelihood ratio function follows the standard chi square distribution asymptotically.Next,we introduce the main results.1.Robust statistical inference of covariate adjusted linear regression model.Suppose that {(Yi,Xi,Ui):i=1,…,n}are independent and identically dis-tributed generated from the following covariate adjusted linear regression model:#12 where ?(=?1,…?p)T is an unknown vector,Ui is an observable covariate,Xi=(Xil,...,Xip)T,Yiand Xi are unobservable variables distorted by smooth function?(Ui)and ?r(Ui).Suppose Ui ?(Yi.XiT).Identifiability constraints for ?(·)and ?r(·),E(?(U))=E(?r(U))=1,such that E(Yi)=E(Yi),E(Xir)=E(Xir).The CAR.model(1)is transformed into a varying-coefficient regression model:Yi=XiT?(Ui)+?(Ui),1?i?n,(2)where ?(Ui)=(?1(Ui),…,?p(Ui))T,?r(Ui)=(?)The robust estimation method is desired.The robust coefficient estimation motivated by Tang and Wang(2005).Two-step estimation procedure is proposed to estimate the unknown parameters.The varying coefficients are estimated by L1 method based on local linear fit.Because model(2)is heteroscedastic,the inferring methods are not same.For Ui in the neighbourhood of u,there is a local linear approximation#12 Let(??,b?)? be the local linear L1-estimate of(??:b?)? by minimizing#12 where a=(a1,…,ap)/T,b=(b1,…bp)T.From identifiability constraints,E(Xr)=E(Xr),E(?r(U)Xr)=?rE(Xr),The unknown regression parameters ?r,r=1,…,p are obtained as averages of raw estimates ?r(Ui).The estimates are given by(?)where(?)Next,we state the consistency result for the estimators ?r Theorem 1 Under the regularity conditions C2.6.1-C2.6.6,if h?0 and nh?? as n??,then#12?? Cn=Op(h2+log1/2(1//h)/(nh)).Theorem 2 Under the regularity conditions C2.6.1-C2.6.6,if nh2/log(1/h)?? and nh4?0 as n??,then the asymptotic distribution of ?r is given by#12 where#12#12 The optimal bandwidth for ?r(·)is h?n-1/5.This bandwidth does not satisfy the condition in Theorem 2.In order to obtain the asymptotic normality for ?r,under-smoothing for ?r(·)is necessary.The requirement has also been used in the literature for semiparametric model;see Carroll et al.(1997)for a detailed discussion.Although we have obtained the asymptotic distribution of ?r,the ??2 is com-plex and includes several unknown components to be estimated.Thus,the empirical likelihood method is proposed to construct a confidence interval for-?r.Note that E((?r(Ui)-?r)Xir)=0,for i=1,2,…,n,r=1,...,p,if ?r is the true parameter.Hence,It is consistence that testing whether ?r is true parameter and testing whether E((?r(Ui)-?r)Xir)=0.By Owen(1991),to construct an empirical likelihood ra-tio function for ?r,we denote V(?r)=(?r(Ui)-?r)Xir.So we define the empirical likelihood ratio function as follows:However,Ln(?r)cannot be directly used to make statistical inference on ?r because Ln(?r)contains the unknown ?r(·).A natural way is to replace ?r(·)by L1-estimator?r(·).Let Vi(?r)be Ui(?r),with ?r(·)replaced by ?r(·),for i=1,2,…,n.Then an estimated empirical likelihood ratio function is defined by(?)In the following,log Ln(?r)converges to the standard chi-square distribution with degree 1.Theorem 3 Under the regularity conditions C2.6.1-C2.6.6,then Ln(?r)(?)?12.Based on Theorem 3,we can construct an(1-?)-level confidence region of ?r CR?={?r:Ln(?r)?c?},where ca satisfies P(?1?c?)=1-?.2.Statistical inference of covariate adjusted partially linear regression model.Suppose that {(Yi,Xi,Ti,Ui),i=1,...,n} are independent and identically dis-tributed generated from the following covariate adjusted partially linear regression model:where g(·)is an unknown smooth function,Xi?=(Xi1,...,Xip)?,Xi?=(Xi1,...,Xip)?.Similar to the discussion in the first part,applying E(?(Ui))=E(?r(Ui))=1,thus#12As is shown by Cui et al.(2009),#12 where(?)is a kernel function,h is a,bandwidth.Let#12The model(6)implies that Yi-E(Yi|Ti)=(Xi-E(Xi|Ti))??+?i.(9)Let g1(t)=E(X1|Ti=t),92(t)=E(Y1|T1=t).When g1(t)and g2(t)are known,model(9)could be considered as a linear model,and then empirical likelihood method could be applied to(9).Let Zi=(Xi-E(Xi|Ti))(Yi-E(Yi|Ti)-(Xi-E(Xi|Ti))??).(10)It is easy to see that E(Zi)=0,when ? is the true parameter.By Owen(1990),the empirical log-likelihood ratio is defined by#12 But,both g1(t)and g2(t)are usually unknown.In addition,Yi,Xi are unobservable variables.To solve the problem,a natural way is to replace Yi and Xi by Yi and Xi respectively.g1(t),g2(t)is replaced by their estimators respectively.Let#12 where Wnj(t)are some non-negative probability weight functions?Wnj(t)can be de-fined by#12 where K*(·)is a kernel function,h*is a bandwidth tending to zero..Let Xi=Xi-g1n(Ti),Yi=Yi-g2n(Ti).Zi in(10)can be replaced by Zi=Xi(Yi-Xi??):1?i?n.The empirical likelihood ratio function can be defined by#12With the assumption that zero is inside the convex hull of the point(Zl,…,Zn),a unique value for Ln(?))exists.Theorem 4 Under the above conditions C3.4.1-C3.4.13,then#12In the following,the theorem show that Ln(?0)converges to the standard chi-square distribution with degree p.Theorem 5 Under the above conditions C3.4.1-C3.4.13,if ?0 is the true value of the parameter,then P(Ln(?0)?c?)=1-?+o(1):where P(?p2<?c?)=1-?.3.Robust statistical inference of linear EV model using instrumental variables.Consider the following linear EV model:where X ? RP,W is the observed surrogate of X.u i is a random measurement error vector,and ? is a random error.The existence of measurement error of independent variable will lead to estimation error.The purpose of research is to correct this de-viation While the instrumental variables method has been widely uses in(13).This method can truly and accurately reflect the relationship between variables.The issue was further studied by Schennach(2007),Abarin and Wang(2012),Xu et al.(2015)and so on.Suppose that there exists a vector of instrumental variables Z that is related to X through X=HZ+v,(14)where H is matrix of unknown parameters.The linear regression(14)of predictive variables and instrumental variables are extended to partial linear regression.X=HZ+g(t)+v.(15)Suppose that {(Yi,Wi,Zi,ti),i=1,...,n} are independent and identically dis-tributed sample from the following model:where Vi=?0Tvi+?i,?i=ui+ui.By Theorem 1 and Theorem 2 of Wang and Zheng(1997),the estimators of H and g(·)in(16)can be obtained.Hence the composite quantile regression estimators for Ck and ?0 can be obtained by minimizing the objective function which is defined as#12????k(r)=?(?k-I(r<0)),K is the number of quantiles.0<?1<?2<…<?K<1.Next,we state the consistency result for the estimators.Theorem 6 Under the above conditions C4.5.1-C4.5.7,when n??,then(?)where?1(k,k')=min(?k,?k')(1-max(?k,?k')),?2(k,k')=-E[fv(ck')?0T?1(I(v1?ck)-?k)],?3(k,k')=-E[fv(ck)?0T?1(I(v1?ck')-?k')],?4(k,k')=E[fv(ck)fv(ck')(?0T?1)2].#12#12However,the limiting variance has a complex structure.The empirical likelihood is applied to(16).Since E{?k-I(Yi-?0?(HZi+g(ti)?Ck)}=0,let#12 It is easy to see that E[?i(?0)]=0.Hence,the problem of testing whether ?0 is true parameter is equivalent to testing whether E[?i(?0)=0.we can define the empirical likelihood ratio function as follows:#12But,H,g(·),ck are usually unknown.To solve the problem,a natural way is to replace H,g(·),Ck by H,g(·),ck respectively.Let#12 Then,the empirical log-likelihood ratio is defined by(?)But,H and g(·)infuence the convergence rate of Ln(?0),such that Ln(?0)is no longer asymptotically subject to the standard chi square distribution.The authors mainly study a corrected empirical likelihood ratio statistic.Let#12 where Kh(·)=h-1K(·/h),K(·)is a kernel function,h is a bandwidth tending to zero.Let If zero is inside the convex set of(?1(?0),…,?n(?0)),a unique value for Ln(?0)exists.Theorem 7 Under the above conditions C4.5.1-C4.5.7,if ?0 is the true value of the parameter,then Ln(?0)(?)?p2;where ?p2 is the standard chi-square distribution with degree p.We construct an(1-?)-level confidence region of ?0 CRa={?0:Ln(?0)?c?},where c? satisfies P(?p2?c?)=1-?.