The paper considers a multivariate partially linear model under independent errors,and investigates the asymptotic bias and variance-covariance for parametric component βand nonparametric component F（·）by the ...The paper considers a multivariate partially linear model under independent errors,and investigates the asymptotic bias and variance-covariance for parametric component βand nonparametric component F（·）by the GJS estimator and Kernel estimation.展开更多
In this paper, we consider the variable selection for the parametric components of varying coefficient partially linear models with censored data. By constructing a penalized auxiliary vector ingeniously, we propose a...In this paper, we consider the variable selection for the parametric components of varying coefficient partially linear models with censored data. By constructing a penalized auxiliary vector ingeniously, we propose an empirical likelihood based variable selection procedure, and show that it is consistent and satisfies the sparsity. The simulation studies show that the proposed variable selection method is workable.展开更多
The paper presents a two-stage method for estimating the parameters in the censored linear model Y-i = max(0, alpha(o) + X-i'beta(o)), 1 less than or equal to i less than or equal to n. In the first stage the data...The paper presents a two-stage method for estimating the parameters in the censored linear model Y-i = max(0, alpha(o) + X-i'beta(o)), 1 less than or equal to i less than or equal to n. In the first stage the data are grouped in some groups and then some adjustments are made, the results are used in the latter stage to form a Tobin-type estimate. The asymptotic normality of the estimate is proved and some simulations are made.展开更多
Variable selection plays an important role in high-dimensional data analysis.But the high-dimensional data often induces the strongly correlated variables problem,which should be properly handled.In this paper,we prop...Variable selection plays an important role in high-dimensional data analysis.But the high-dimensional data often induces the strongly correlated variables problem,which should be properly handled.In this paper,we propose Elastic Net procedure for partially linear models and prove the group effect of its estimate.A simulation study shows that the Elastic Net procedure deals with the strongly correlated variables problem better than the Lasso,ALasso and the Ridge do.Based on the real world data study,we can get that the Elastic Net procedure is particularly useful when the number of predictors pffis much bigger than the sample size n.展开更多
An empirical likelihood approach to estimate the coefficients in linear model with interval censored responses is developed in this paper. By constructing unbiased transformation of interval censored data,an empirical...An empirical likelihood approach to estimate the coefficients in linear model with interval censored responses is developed in this paper. By constructing unbiased transformation of interval censored data,an empirical log-likelihood function with asymptotic X^2 is derived. The confidence regions for the coefficients are constructed. Some simulation results indicate that the method performs better than the normal approximation method in term of coverage accuracies.展开更多
The 3-hour-interval prediction of ground-level temperature from +00 h out to +45 h in South Korea (38 stations) is performed using the DLM (dynamic linear model) in order to eliminate the systematic error of numerical...The 3-hour-interval prediction of ground-level temperature from +00 h out to +45 h in South Korea (38 stations) is performed using the DLM (dynamic linear model) in order to eliminate the systematic error of numerical model forecasts. Numerical model forecasts and observations are used as input values of the DLM. According to the comparison of the DLM forecasts to the KFM (Kalman filter model) forecasts with RMSE and bias, the DLM is useful to improve the accuracy of prediction.展开更多
In this paper,the empirical likelihood confidence regions for the regression coefficient in a linear model are constructed under m-dependent errors.It is shown that the blockwise empirical likelihood is a good way to ...In this paper,the empirical likelihood confidence regions for the regression coefficient in a linear model are constructed under m-dependent errors.It is shown that the blockwise empirical likelihood is a good way to deal with dependent samples.展开更多
Consider the semiparametric varying-coefficient heteroscedastic partially linear model Yi = X^T i β＋ Z^T iα（Ti） ＋ σiei, 1 ≤ i≤ n, where σ ^2i= f（Ui）, β is a p × 1 column vector of unknown parameter, ...Consider the semiparametric varying-coefficient heteroscedastic partially linear model Yi = X^T i β＋ Z^T iα（Ti） ＋ σiei, 1 ≤ i≤ n, where σ ^2i= f（Ui）, β is a p × 1 column vector of unknown parameter, （Xi, Zi, Ti, Ui） are random design q-dimensional vector of unknown functions, el points, Yi are the response variables, α（-） is a are random errors. For both cases that f（.） is known and unknown, we propose the empirical log-likelihood ratio statistics for the parameter f（.）. For each case, a nonparametric version of Wilks＇ theorem is derived. The results are then used to construct confidence regions of the parameter. Simulation studies are carried out to assess the performance of the empirical likelihood method.展开更多
This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） prop...This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） proposed a profile least squares estimator for the parametric component and established its asymptotic normality. We further show that the profile least squares estimator can achieve the law of iterated logarithm. Moreover, we study the estimators of the functions characterizing the non-linear part as well as the error variance. The strong convergence rate and the law of iterated logarithm are derived for them, respectively.展开更多
The present paper daisses the relative efficiencies of the least square estimates in linear models. For Gauss-Markoff model: Y=Xe + e E(e)= 0, Cov(e)=V, an new efficiencyo f least square estimate for linearly estimabl...The present paper daisses the relative efficiencies of the least square estimates in linear models. For Gauss-Markoff model: Y=Xe + e E(e)= 0, Cov(e)=V, an new efficiencyo f least square estimate for linearly estimable function c'r is proposed and its lower bound is giv-en. For variance component model: Y=X + e, E(e)=0, Cov(e)=, an new efficiency of least square estimate for linearly estimable function C'r is introduced for the first timeand its lower bound, which is independent of unknown parameters, is also obtained.展开更多
This paper presents a study on the statistical forecasts of typhoon tracks. Numerical models have their own systematic errors, like a bias. In order to improve the accuracy of track forecasting, a statistical model ca...This paper presents a study on the statistical forecasts of typhoon tracks. Numerical models have their own systematic errors, like a bias. In order to improve the accuracy of track forecasting, a statistical model called DLM (dynamic linear model) is applied to remove the systematic error. In the analysis of typhoons occurring over the western North Pacific in 1997 and 2000, DLM is useful as an adaptive model for the prediction of typhoon tracks.展开更多
This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allo...This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.展开更多
We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alte...We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.展开更多
We study the quasi likelihood equation in Generalized Linear Models(GLM)with adaptive design∑(i=1)^n xi(yi-h(x'iβ))=0,where yi is a q=vector,and xi is a p×q random matrix.Under some assumptions,it is shown ...We study the quasi likelihood equation in Generalized Linear Models(GLM)with adaptive design∑(i=1)^n xi(yi-h(x'iβ))=0,where yi is a q=vector,and xi is a p×q random matrix.Under some assumptions,it is shown that the Quasi-Likelihood equation for the GLM has a solution which is asymptotic normal.展开更多
Regression models are often transformed into certain alternative forms in statistical inference theory.In this paper,we assume that a general linear model(GLM)is transformed into two diferent forms,and our aim is to s...Regression models are often transformed into certain alternative forms in statistical inference theory.In this paper,we assume that a general linear model(GLM)is transformed into two diferent forms,and our aim is to study some comparison problems under the two transformed general linear models(TGLMs).We frst construct a general vector composed of all unknown parameters under the two diferent TGLMs,derive exact expressions of best linear minimum bias predictors(BLMBPs)by solving a constrained quadratic matrix-valued function optimization problem in the L¨owner partial ordering,and describe a variety of mathematical and statistical properties and performances of the BLMBPs.We then approach some algebraic characterization problems concerning relationships between the BLMBPs under two diferent TGLMs.As applications,two specifc cases are presented to illustrate the main contributions in the study.展开更多
Multivariate regression models have been extensively studied in the literature and applied in practice.It is not unusual that some predictors may make the same nonnull contributions to all the elements of the response...Multivariate regression models have been extensively studied in the literature and applied in practice.It is not unusual that some predictors may make the same nonnull contributions to all the elements of the response vector,especially when the number of predictors is very large.For convenience,we call the set of such predictors as the homogeneity set.In this paper,we consider a sparse high-dimensional multivariate generalized linear models with coexisting homogeneity and heterogeneity sets of predictors,which is very important to facilitate the understanding of the effects of different types of predictors as well as improvement on the estimation efficiency.We propose a novel adaptive regularized method by which we can easily identify the homogeneity set of predictors and investigate the asymptotic properties of the parameter estimation.More importantly,the proposed method yields a smaller variance for parameter estimation compared to the ones that do not consider the existence of a homogeneity set of predictors.We also provide a computational algorithm and present its theoretical justification.In addition,we perform extensive simulation studies and present real data examples to demonstrate the proposed method.展开更多
Under the assumption that in the generalized linear model (GLM) the expectation of the response variable has a correct specification and some other smooth conditions, it is shown that with probability one the quasi-li...Under the assumption that in the generalized linear model (GLM) the expectation of the response variable has a correct specification and some other smooth conditions, it is shown that with probability one the quasi-likelihood equation for the GLM has a solution when the sample size n is sufficiently large. The rate of this solution tending to the true value is determined. In an important special case, this rate is the same as specified in the LIL for iid partial sums and thus cannot be improved anymore.展开更多
In this paper, we explore some weakly consistent properties of quasi-maximum likelihood estimates (QMLE) concerning the quasi-likelihood equation $ \sum\nolimits_{i = 1}^n {X_i (y_i - \mu (X_i^\prime \beta ))} $ for u...In this paper, we explore some weakly consistent properties of quasi-maximum likelihood estimates (QMLE) concerning the quasi-likelihood equation $ \sum\nolimits_{i = 1}^n {X_i (y_i - \mu (X_i^\prime \beta ))} $ for univariate generalized linear model E(y|X) = μ(X′β). Given uncorrelated residuals {e i = Y i ? μ(X i ′ β0), 1 ? i ? n} and other conditions, we prove that $$ \hat \beta _n - \beta _0 = O_p (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n^{ - 1/2} ) $$ holds, where $ \hat \beta _n $ is a root of the above equation, β 0 is the true value of parameter β and $$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n $$ denotes the smallest eigenvalue of the matrix S n = ∑ i=1 n X i X i ′ . We also show that the convergence rate above is sharp, provided independent non-asymptotically degenerate residual sequence and other conditions. Moreover, paralleling to the elegant result of Drygas (1976) for classical linear regression models, we point out that the necessary condition guaranteeing the weak consistency of QMLE is S n ?1 → 0, as the sample size n → ∞.展开更多
For generalized linear models （GLM）, in case the regressors are stochastic and have different distributions, the asymptotic properties of the maximum likelihood estimate （MLE） β^n of the parameters are studied. U...For generalized linear models （GLM）, in case the regressors are stochastic and have different distributions, the asymptotic properties of the maximum likelihood estimate （MLE） β^n of the parameters are studied. Under reasonable conditions, we prove the weak, strong consistency and asymptotic normality of β^n展开更多
We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a noncon- cave regul...We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a noncon- cave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o（√n）, where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance. Comprehensive simulation studies are carried out and an application is presented to examine the finite-sample performance of the proposed procedures.展开更多
基金Supported by the Anhui Provincial Natural Science Foundation（11040606M04） Supported by the National Natural Science Foundation of China（10871001,10971097）
文摘The paper considers a multivariate partially linear model under independent errors,and investigates the asymptotic bias and variance-covariance for parametric component βand nonparametric component F（·）by the GJS estimator and Kernel estimation.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1110111911126332)+2 种基金the National Social Science Foundation of China(Grant No.11CTJ004)the Natural Science Foundation of Guangxi Province(Grant No.2010GXNSFB013051)the Philosophy and Social Sciences Foundation of Guangxi Province(Grant No.11FTJ002)
文摘In this paper, we consider the variable selection for the parametric components of varying coefficient partially linear models with censored data. By constructing a penalized auxiliary vector ingeniously, we propose an empirical likelihood based variable selection procedure, and show that it is consistent and satisfies the sparsity. The simulation studies show that the proposed variable selection method is workable.
文摘The paper presents a two-stage method for estimating the parameters in the censored linear model Y-i = max(0, alpha(o) + X-i'beta(o)), 1 less than or equal to i less than or equal to n. In the first stage the data are grouped in some groups and then some adjustments are made, the results are used in the latter stage to form a Tobin-type estimate. The asymptotic normality of the estimate is proved and some simulations are made.
基金Supported by National Natural Science Foundation of China(No.71462002)the Project for Teaching Reform of Guangxi(GXZZJG2017B084)the Project for Fostering Distinguished Youth Scholars of Guangxi(2020KY50012)。
文摘Variable selection plays an important role in high-dimensional data analysis.But the high-dimensional data often induces the strongly correlated variables problem,which should be properly handled.In this paper,we propose Elastic Net procedure for partially linear models and prove the group effect of its estimate.A simulation study shows that the Elastic Net procedure deals with the strongly correlated variables problem better than the Lasso,ALasso and the Ridge do.Based on the real world data study,we can get that the Elastic Net procedure is particularly useful when the number of predictors pffis much bigger than the sample size n.
文摘An empirical likelihood approach to estimate the coefficients in linear model with interval censored responses is developed in this paper. By constructing unbiased transformation of interval censored data,an empirical log-likelihood function with asymptotic X^2 is derived. The confidence regions for the coefficients are constructed. Some simulation results indicate that the method performs better than the normal approximation method in term of coverage accuracies.
文摘The 3-hour-interval prediction of ground-level temperature from +00 h out to +45 h in South Korea (38 stations) is performed using the DLM (dynamic linear model) in order to eliminate the systematic error of numerical model forecasts. Numerical model forecasts and observations are used as input values of the DLM. According to the comparison of the DLM forecasts to the KFM (Kalman filter model) forecasts with RMSE and bias, the DLM is useful to improve the accuracy of prediction.
文摘In this paper,the empirical likelihood confidence regions for the regression coefficient in a linear model are constructed under m-dependent errors.It is shown that the blockwise empirical likelihood is a good way to deal with dependent samples.
基金Supported by the National Natural Science Foundation of China (Grant No. 71171003)Natural Science Research Project of Anhui Provincial Colleges (Grant No. KJ2011A032)+3 种基金Anhui Polytechnic University Foundation for Recruiting Talent (Grant Nos. 2011YQQ0042009YQQ005)Young Teachers Science Research Foundation of Anhui Polytechnic University (Grant No. 2009YQ035)Anhui Provincial Natural Science Foundation
文摘Consider the semiparametric varying-coefficient heteroscedastic partially linear model Yi = X^T i β＋ Z^T iα（Ti） ＋ σiei, 1 ≤ i≤ n, where σ ^2i= f（Ui）, β is a p × 1 column vector of unknown parameter, （Xi, Zi, Ti, Ui） are random design q-dimensional vector of unknown functions, el points, Yi are the response variables, α（-） is a are random errors. For both cases that f（.） is known and unknown, we propose the empirical log-likelihood ratio statistics for the parameter f（.）. For each case, a nonparametric version of Wilks＇ theorem is derived. The results are then used to construct confidence regions of the parameter. Simulation studies are carried out to assess the performance of the empirical likelihood method.
基金supported by the National Natural Science Funds for Distinguished Young Scholar (70825004)National Natural Science Foundation of China (NSFC) (10731010 and 10628104)+3 种基金the National Basic Research Program (2007CB814902)Creative Research Groups of China (10721101)Leading Academic Discipline Program, the 10th five year plan of 211 Project for Shanghai University of Finance and Economics211 Project for Shanghai University of Financeand Economics (the 3rd phase)
文摘This article is concerned with the estimating problem of semiparametric varyingcoefficient partially linear regression models. By combining the local polynomial and least squares procedures Fan and Huang （2005） proposed a profile least squares estimator for the parametric component and established its asymptotic normality. We further show that the profile least squares estimator can achieve the law of iterated logarithm. Moreover, we study the estimators of the functions characterizing the non-linear part as well as the error variance. The strong convergence rate and the law of iterated logarithm are derived for them, respectively.
文摘The present paper daisses the relative efficiencies of the least square estimates in linear models. For Gauss-Markoff model: Y=Xe + e E(e)= 0, Cov(e)=V, an new efficiencyo f least square estimate for linearly estimable function c'r is proposed and its lower bound is giv-en. For variance component model: Y=X + e, E(e)=0, Cov(e)=, an new efficiency of least square estimate for linearly estimable function C'r is introduced for the first timeand its lower bound, which is independent of unknown parameters, is also obtained.
基金the project"A study on improving forecast skill using a su-percomputer"of Meteorological Research Institute,KMA,2001.
文摘This paper presents a study on the statistical forecasts of typhoon tracks. Numerical models have their own systematic errors, like a bias. In order to improve the accuracy of track forecasting, a statistical model called DLM (dynamic linear model) is applied to remove the systematic error. In the analysis of typhoons occurring over the western North Pacific in 1997 and 2000, DLM is useful as an adaptive model for the prediction of typhoon tracks.
基金The talent research fund launched (3004-893325) of Dalian University of Technologythe NNSF (10271049) of China.
文摘This article concerded with a semiparametric generalized partial linear model （GPLM） with the type Ⅱ censored data. A sieve maximum likelihood estimator （MLE） is proposed to estimate the parameter component, allowing exploration of the nonlinear relationship between a certain covariate and the response function. Asymptotic properties of the proposed sieve MLEs are discussed. Under some mild conditions, the estimators are shown to be strongly consistent. Moreover, the estimators of the unknown parameters are asymptotically normal and efficient, and the estimator of the nonparametric function has an optimal convergence rate.
文摘We propose the test statistic to check whether the nonpararnetric functions in two partially linear models are equality or not in this paper. We estimate the nonparametric function both in null hypothesis and the alternative by the local linear method, where we ignore the parametric components, and then estimate the parameters by the two stage method. The test statistic is derived, and it is shown to be asymptotically normal under the null hypothesis.
基金Supported by the National Natural Science Foundation of China(10371092)
文摘We study the quasi likelihood equation in Generalized Linear Models(GLM)with adaptive design∑(i=1)^n xi(yi-h(x'iβ))=0,where yi is a q=vector,and xi is a p×q random matrix.Under some assumptions,it is shown that the Quasi-Likelihood equation for the GLM has a solution which is asymptotic normal.
文摘Regression models are often transformed into certain alternative forms in statistical inference theory.In this paper,we assume that a general linear model(GLM)is transformed into two diferent forms,and our aim is to study some comparison problems under the two transformed general linear models(TGLMs).We frst construct a general vector composed of all unknown parameters under the two diferent TGLMs,derive exact expressions of best linear minimum bias predictors(BLMBPs)by solving a constrained quadratic matrix-valued function optimization problem in the L¨owner partial ordering,and describe a variety of mathematical and statistical properties and performances of the BLMBPs.We then approach some algebraic characterization problems concerning relationships between the BLMBPs under two diferent TGLMs.As applications,two specifc cases are presented to illustrate the main contributions in the study.
文摘Multivariate regression models have been extensively studied in the literature and applied in practice.It is not unusual that some predictors may make the same nonnull contributions to all the elements of the response vector,especially when the number of predictors is very large.For convenience,we call the set of such predictors as the homogeneity set.In this paper,we consider a sparse high-dimensional multivariate generalized linear models with coexisting homogeneity and heterogeneity sets of predictors,which is very important to facilitate the understanding of the effects of different types of predictors as well as improvement on the estimation efficiency.We propose a novel adaptive regularized method by which we can easily identify the homogeneity set of predictors and investigate the asymptotic properties of the parameter estimation.More importantly,the proposed method yields a smaller variance for parameter estimation compared to the ones that do not consider the existence of a homogeneity set of predictors.We also provide a computational algorithm and present its theoretical justification.In addition,we perform extensive simulation studies and present real data examples to demonstrate the proposed method.
基金This work was supported by the National Natural Science Foundation of China.
文摘Under the assumption that in the generalized linear model (GLM) the expectation of the response variable has a correct specification and some other smooth conditions, it is shown that with probability one the quasi-likelihood equation for the GLM has a solution when the sample size n is sufficiently large. The rate of this solution tending to the true value is determined. In an important special case, this rate is the same as specified in the LIL for iid partial sums and thus cannot be improved anymore.
基金supported by the President Foundation (Grant No. Y1050)the Scientific Research Foundation(Grant No. KYQD200502) of GUCAS
文摘In this paper, we explore some weakly consistent properties of quasi-maximum likelihood estimates (QMLE) concerning the quasi-likelihood equation $ \sum\nolimits_{i = 1}^n {X_i (y_i - \mu (X_i^\prime \beta ))} $ for univariate generalized linear model E(y|X) = μ(X′β). Given uncorrelated residuals {e i = Y i ? μ(X i ′ β0), 1 ? i ? n} and other conditions, we prove that $$ \hat \beta _n - \beta _0 = O_p (\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n^{ - 1/2} ) $$ holds, where $ \hat \beta _n $ is a root of the above equation, β 0 is the true value of parameter β and $$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } _n $$ denotes the smallest eigenvalue of the matrix S n = ∑ i=1 n X i X i ′ . We also show that the convergence rate above is sharp, provided independent non-asymptotically degenerate residual sequence and other conditions. Moreover, paralleling to the elegant result of Drygas (1976) for classical linear regression models, we point out that the necessary condition guaranteeing the weak consistency of QMLE is S n ?1 → 0, as the sample size n → ∞.
基金Project supported by the Chinese Natural Science Foundation
文摘For generalized linear models （GLM）, in case the regressors are stochastic and have different distributions, the asymptotic properties of the maximum likelihood estimate （MLE） β^n of the parameters are studied. Under reasonable conditions, we prove the weak, strong consistency and asymptotic normality of β^n
基金supported by National Institute on Drug Abuse(Grant Nos.R21-DA024260 and P50-DA10075)National Natural Science Foundation of China(Grant Nos.11071077,11371236,11028103,11071022 and 11028103)+2 种基金Innovation Program of Shanghai Municipal Education CommissionPujiang Project of Science and Technology Commission of Shanghai Municipality(Grant No.12PJ1403200)Program for New Century Excellent Talents,Ministry of Education of China(Grant No.NCET-12-0901)
文摘We are concerned with robust estimation procedures to estimate the parameters in partially linear models with large-dimensional covariates. To enhance the interpretability, we suggest implementing a noncon- cave regularization method in the robust estimation procedure to select important covariates from the linear component. We establish the consistency for both the linear and the nonlinear components when the covariate dimension diverges at the rate of o（√n）, where n is the sample size. We show that the robust estimate of linear component performs asymptotically as well as its oracle counterpart which assumes the baseline function and the unimportant covariates were known a priori. With a consistent estimator of the linear component, we estimate the nonparametric component by a robust local linear regression. It is proved that the robust estimate of nonlinear component performs asymptotically as well as if the linear component were known in advance. Comprehensive simulation studies are carried out and an application is presented to examine the finite-sample performance of the proposed procedures.
