Estimation of Partial Linear Error-in-Variables Models under Martingale Difference Sequence
Received:April 01, 2014  Revised:June 18, 2014
Key Words: partial linear error-in-variables models   martingale difference sequence   validation data   strong consistency
Fund Project:Supported by National Natural Science Foundation of China (Grant Nos.11271155; 11371168; 11001105; 11071126; 11071269), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20110061110003), the Natural Science Foundation of Jilin Province (Grant Nos.20130101066JC; 20130522102JH; 20101596), Twelfth Five-Year Plan' Science and Technology Research Project of the Education Department of Jilin Province (Grant No.2012186).
 Author Name Affiliation Zhuoxi YU School of Management Science and Information Engineering, Jilin University of Finance and Economics, Jilin 130117, P. R. China Dehui WANG Department of Statistic, College of Mathematics, Jilin University, Jilin 130021, P. R. China Na HUANG School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, P. R. China School of Management Science and Information Engineering, Jilin University of Finance and Economics, Jilin 130117, P. R. China
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Consider the partly linear model $Y=x\beta+g(t)+e$ where the explanatory $x$ is erroneously measured, and both $t$ and the response $Y$ are measured exactly, the random error $e$ is a martingale difference sequence. Let $\widetilde{x}$ be a surrogate variable observed instead of the true $x$ in the primary survey data. Assume that in addition to the primary data set containing $N$ observations of $\{(Y_{j},\widetilde{x}_{j},t_{j})_{j=n+1}^{n+N}\}$, the independent validation data containing $n$ observations of $\{(\widetilde{x}_{j},x_{j},t_{j})_{j=1}^{n}\}$ is available. In this paper, a semiparametric method with the primary data is employed to obtain the estimator of $\beta$ and $g(\cdot)$ based on the least squares criterion with the help of validation data. The proposed estimators are proved to be strongly consistent. Finite sample behavior of the estimators is investigated via simulations too.