Abstract
This paper is concerned with the estimating problem of a semiparametric varying-coefficient partially linear errors-in-variables model Y i = X t i β + Z t i α(U i ) + ɛ i , W i = X i + ζ i , i = 1, …, n. Due to measurement errors, the usual profile least square estimator of the parametric component, local polynomial estimator of the nonparametric component and profile least squares based estimator of the error variance are biased and inconsistent. By taking the measurement errors into account we propose a generalized profile least squares estimator for the parametric component and show it is consistent and asymptotically normal. Correspondingly, the consistent estimation of the nonparametric component and error variance are proposed as well. These results may be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of these proposed estimations.
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References
Cai Z., Fan J., Yao, Q. Functional-coefficient regression models for nonlinear time series. J. Amer. Statist. Assoc., 95: 941–956 (2000)
Cai Z., Fan, J., Li, R. Efficient estimation and inferences for varying-coefficient models. J. Amer. Statist. Assoc., 95: 888–902 (2000)
Carroll R.J., Ruppert, D., Stefanski, L.A. Measurement Error in Nonlinear Models. Chapman & Hall, 1995
Chow, Y.S., Teicher, H. Probability theory. Independence, interchangeability, martingales. Springer-Verlag, New York, Heidelberg, 1978
Cui, H.J., Li R.C. On parameter estimation for semi-linear errors-in-variables models. J. Mult. Anal., 64: 1–24 (1998)
Fan, J., Gijbels, I. Data driven bandwidth slection in local polynomial fitting: Variable bandwidth spatial adaptation. J. Royal. Stat. Soc., B, 30: 238–348 (1995)
Fan, J., Huang, T. Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 11: 1031–1057 (2005)
Li Q., Huang C.J., Li, D., Fu, T.T. Semiparametric smooth coefficient models. J. Business and Econ. Statist., 20: 412–422 (2002)
Liang, H., Thurston, S., Ruppert, D., Apanasovich, T., Hauser, R. Additive partial linear models with measurement errors. Biometrika, 95: 667–678 (2008)
Liang, H., Wang, S.J., Carroll, R. Partially linear models with missing response variables and error-prone covariates. Biometrika, 94: 185–198 (2007)
Liang, H., Wang, S.J., Robins, J., Carroll, R. Estimation in partially linear models with missing covariates. J. Amer. Statist. Assoc., 99: 357–367 (2004)
Ruppert, D., Sheather, S.J., Wand, M.P. An effective bandwidth selector for local least squares regression. J. Amer. Statist. Assoc., 90: 1257–1270 (1995)
Tosteson, T., Stefanski, L.A., Schafer, D.W. A measurement error model for binary and ordinal regression. Statist. Medicine, 8: 1139–1147 (1999)
Xia, Y, Li, W.K. On the estimation and testing of functional-coefficient linear models. Statist. Sinica, 9: 737–757 (1999)
Wang, N., Carroll, R.J., Liang, K.Y. Quasilikelihood and variance functions in measurement error models with replicates. Biometrics, 52: 423–432 (1996)
Wang, X.J. Statistical analysis on on semiparametric varying-coefficient partially linear models with incomplete data. Master paper, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 2008
You, J.H., Zhou, X. Wavelet estimation in varying-coefficient partially linear regression models. Statist. Prob. Lett., 68: 91–104 (2004)
You, J.H., Chen, G. Estimation in a semiparametric varying-coefficient partially linear errors-in-variables model. J. Multi. Anal., 97: 324–341 (2006)
Zhang, W., Lee, S.Y., Song, X. Local polynomial fitting in semivarying coefficient models. J. Multi. Anal., 82: 166–188 (2002)
Zhou, Y., Liang, H. Statistical inference for semiparametric varying-coefficient partially linear models with generated regressors. Ann. Statist., 37: 427–458 (2009)
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Zhou’s work was partially supported by National Natural Science Funds for Distinguished Young Scholar (No.70825004) and (No.71271128), Creative Research Groups of China (No.71271128), NCMIS and Shanghai University of Finance and Economics through Project 211 Phase III and Shanghai Leading Academic Discipline Project (No. B803).
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Ma, Yb., You, Jh. & Zhou, Y. Generalized profile LSE in varying-coefficient partially linear models with measurement errors. Acta Math. Appl. Sin. Engl. Ser. 29, 477–490 (2013). https://doi.org/10.1007/s10255-013-0229-z
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DOI: https://doi.org/10.1007/s10255-013-0229-z
Keywords
- Semiparametric modeling
- varying-coefficient
- measurement error
- local polynomial
- profile least squares
- asymptotic normality