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Asymptotic normality of estimators in heteroscedastic errors-in-variables model

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Abstract

This article is concerned with the estimating problem of heteroscedastic partially linear errors-in-variables models. We derive the asymptotic normality for estimators of the slope parameter and the nonparametric component in the case of known error variance with stationary \(\alpha \)-mixing random errors. Also, when the error variance is unknown, the asymptotic normality for the estimators of the slope parameter and the nonparametric component as well as variance function is considered under independent assumptions. Finite sample behavior of the estimators is investigated via simulations too.

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References

  • Aneiros-Pérez, G., González-Manteiga, W., Vieu, P.: Estimation and testing in a partial linear regression model under long-memory dependence. Bernoulli 10, 49–78 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Carroll, R.J., Ruppert, D., Stefanski, L.A.: Measurement Error in Nonlinear Models. Chapman Hall, London (1995)

    Book  MATH  Google Scholar 

  • Cheng, C.L., Tsai, C.L.: The invariance of some score test in the linear measurement error model. J. Am. Stat. Assoc. 99, 805–809 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  • Cheng, C.L., Van Ness, J.W.: Statistical Regression with Measurement Error. Arnold, London (1999)

    MATH  Google Scholar 

  • Cui, H., Li, R.: On parameter estimation for semi-linear errors-in-variables models. J. Multivar. Anal. 64, 1–24 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  • Doukhan, P.: Mixing properties and examples. Lecture Notes in Statistics, vol. 85. Springer, Berlin (1994)

  • Engle, R., Granger, C., Rice, J., Weiss, A.: Nonparametric estimates of the relation between weather and electricity sales. J. Am. Stat. Assoc. 81, 310–320 (1986)

    Article  Google Scholar 

  • Fan, G.L., Liang, H.Y., Wang, J.F., Xu, H.X.: Asymptotic properties for LS estimators in EV regression model with dependent errors. AStA Adv. Stat. Anal. 94, 89–103 (2010)

    Article  MathSciNet  Google Scholar 

  • Fuller, W.A.: Measurement Error Models. Wiley, New York (1987)

    Book  MATH  Google Scholar 

  • Gao, J.T., Chen, X.R., Zhao, L.C.: Asymptotic normality of a class of estimators in partial linear models. Acta. Math. Sinica 37(2), 256–268 (1994)

    MATH  MathSciNet  Google Scholar 

  • Hall, P., Heyde, C.C.: Martingale Limit Theory and its Applications. Academic Press, New York (1980)

    Google Scholar 

  • Hamilton, S.A., Truong, Y.K.: Local linear estimation in partly linear models. J. Multivar. Anal. 60, 1–19 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  • Härdle, W., Liang, H., Gao, J.: Partial Linear Models. Physica-Verlag, Heidelberg (2000)

    Book  Google Scholar 

  • Liang, H., Härdle, W., Carroll, R.J.: Estimation in a semiparametric partially linear errors-in-variables model. Ann. Stat. 27, 1519–1535 (1999)

    Article  MATH  Google Scholar 

  • Liang, H.Y., Jing, B.Y.: Asympototic normality in partial linear models based on depengdent errors. J. Stat. Plann. Inference 139, 1357–1371 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Liu, J.X., Chen, X.R.: Consistency of LS estimator in simple linear EV regression models. Acta Math. Sci. Ser. B 25, 50–58 (2005)

    MATH  Google Scholar 

  • Miao, Y., Liu, W.: Moderate deviations for LS estimator in simple linear EV regression model. J. Stat. Plann. Inference 139(9), 3122–3131 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  • Miao, Y., Yang, G., Shen, L.: The central limit theorem for LS estimator in simple linear EV regression models. Comm. Stat. Theory Methods 36, 2263–2272 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • Patriota, A.G., Lemonte, A.J., Bolfarine, H.: Improved maximum likelihood estimators in a heteroscedastic errors-in-variables model. Stat. Papers 52, 455–467 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Volkonskii, V.A., Rozanov, Y.A.: Some limit theorems for random functions. Theory Probab. Appl. 4, 178–197 (1959)

    Article  MathSciNet  Google Scholar 

  • Withers, C.S.: Conditions for linear processes to be strong mixing. Z. Wahrsch. Verw. Gebiete 57, 477–480 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  • Yang, S.C.: Maximal moment inequality for partial sums of strong mixing sequences and application. Act. Math. Sinica (English Series) 23(6), 1013–1024 (2007)

    Google Scholar 

  • You, J., Chen, G., Zhou, Y.: Statistical inference of partially linear regression models with heteroscedastic errors. J. Multivar. Anal. 98, 1539–1557 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  • You, J.H., Zhou, X., Zhou, Y.: Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors. J. Multivar. Anal. 101, 1079–1101 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  • Zhang, J.J., Liang, H.Y.: Berry-Esseen type bounds in heteroscedastic semi-parametric model. J. Stat. Plann. Inference 141, 3447–3462 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  • Zhou, H.B., You, J.H., Zhou, B.: Statistical inference for fixed-effects partially linear regression models with errors in variables. Stat. Papers 51, 629–650 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The authors were supported by the National Natural Science Foundation of China (11271286) and the Specialized Research Fund for the Doctor Program of Higher Education of China (20120072110007).

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, People’s Republic of China

    Jing-Jing Zhang

  2. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Han-Ying Liang

  3. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154, USA

    Amei Amei

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  1. Jing-Jing Zhang
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  2. Han-Ying Liang
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Correspondence to Han-Ying Liang.

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Zhang, JJ., Liang, HY. & Amei, A. Asymptotic normality of estimators in heteroscedastic errors-in-variables model. AStA Adv Stat Anal 98, 165–195 (2014). https://doi.org/10.1007/s10182-013-0224-y

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  • DOI: https://doi.org/10.1007/s10182-013-0224-y

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