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Robust Nonparametric Function Estimation for Errors-in-variables Models

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Abstract

This paper discusses robust nonparametric estimators of location regression function for errors-in-variables model with de-convolution kernel. The local constant smoother is used for the estimation of the nonparametric function, and the local linear smoother is proposed to deal with the boundary problem, as well as to improve the local constant smoother. We establish the asymptotic properties of the estimator, the in uence function of the statistical functional and the breakdown point. A simulation study is carried out to demonstrate robust performance of the proposed estimator. The motorcycle data is presented to illustrate the application of the robust estimator further.

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References

  1. Azzalini, A., Bowman, A. W., Hardle, W. Robust estimation in the errors-invariables model. Biometrika, 76(1): 149–160 (1989)

    Article  MathSciNet  Google Scholar 

  2. Carroll, R. J., Hall, P. Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association, 83(404): 1184–1186 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Carroll, R.J., Maca, J.D., Ruppert, D. Nonparametric regression in the presence of measurement error. Biometrika, 86(3): 541–554 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carroll, R.J., Ruppert, D., Stefanski, L.A. Measurement Error in Nonlinear Models. Chapman & Hall, Boca Raton, 1995

    Book  MATH  Google Scholar 

  5. Cui, H.J. Asymptotic properties of generalized mad estimators in EV model. Sci. China, 2: 119–131 (1997)

    Google Scholar 

  6. Cui, H.J., Kong, E.F. Empirical likelihood confidence region for parameters in semi-linear errors-in-variables models. Scandinavian Journal of Statistics, 33(1): 153–168 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fan, J.Q. Asymptotic normality for deconvolution kernel density estimators. Sankhy: The Indian Journal of Statistics (Series A, 1961-2002), 53(1): 97–110 (1991)

    MATH  Google Scholar 

  8. Fan, J.Q., Hu, T.C., Truong, Y.K. Robust non-parametric function estimation. Scandinavian Journal of Statistics, 21(4): 433–446 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Fan, J.Q., Truong, Y.K. Nonparametric regression with errors in variables. Annals of Statistics, 21(4): 1900–1925 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, J.Q., Gijbels, R. Local Polynomial Modelling and Its Applications. Chapman & Hall, Boca Raton, 1996

    MATH  Google Scholar 

  11. Wayne, A. Fuller. Measurement Error Models. Wiley, New Jersey, 1987

    MATH  Google Scholar 

  12. Hu, H.C., Cui, H.J., Li, K.C. Asymptotic properties of wavelet estimators in partially linear errors-in-variables models with long-memory errors. Acta Mathematicae Applicatae Sinica (English Series), 34(1): 77–96 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hu, T., Cui, H.J. T-type estimators for a class of linear errors in variables models. Statistica Sinica, 19(3): 1013–1036 (2009)

    MathSciNet  MATH  Google Scholar 

  14. Huber, P. Robust Statistics. Wiley Interscience, New Jersey, 2011

    Google Scholar 

  15. Jin, J., Zhu, L., Tong, X., Ness, K.K. T-type corrected-loss estimation for errorin-variable model. Com- munications in Statistics Theory & Methods, 46(2): 616–627 (2017)

    Article  MATH  Google Scholar 

  16. Kang, M.J. Least trimmed squares estimator in the errors-in-variables model. Journal of Applied Statistics, 34(3): 331–338 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, G.R., Zhang J., Feng, S.Y. Modern measurement error model. Science Press, Beijing, 2016

    Google Scholar 

  18. Li, T. Robust and consistent estimation of nonlinear errors-in-variables models. Journal of Econometrics, 110(1): 1–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pepe, M.S., Fleming, T.R. A nonparametric method for dealing with mismeasured covariate data. Journal of the American Statistical Association, 86(413): 108–113 (1991)

    Article  MathSciNet  Google Scholar 

  20. Stefanski, L.A., Carroll, R.J. Deconvoluting kernel density estimators. Statistics, 21(2): 169–184 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stefanski, L.A. Rates of convergence of some estimators in a class of deconvolution problems. Statistics & Probability Letters, 9(3): 229–235 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  22. Taylor, R.L. Zhang, H.M. On a strongly consistent nonparametric density estimator for the deconvolution problem. Communications in Statistics, 19(9): 3325–3342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhu, L.X., Cui, H.J. A semi-parametric regression model with errors in variables. Scandinavian Journal of Statistics, 30(2): 429–442 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We would like to thank the anonymous referees for their valuable comments and suggestions that helped us to improve the manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Chao-xia Yuan & Heng-jian Cui

Authors
  1. Chao-xia Yuan
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  2. Heng-jian Cui
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Correspondence to Heng-jian Cui.

Additional information

This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 1971324, 11471223), Capacity Building for Sci-Tech Innovation-Fundamental Scientific Research Funds (No:19530050181), Interdiscipline for Bioinformatics and Statistics and Academy for Multidisciplinary Studies of Capital Normal University, Beijing.

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Yuan, Cx., Cui, Hj. Robust Nonparametric Function Estimation for Errors-in-variables Models. Acta Math. Appl. Sin. Engl. Ser. 36, 314–331 (2020). https://doi.org/10.1007/s10255-020-0944-1

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  • DOI: https://doi.org/10.1007/s10255-020-0944-1

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