Estimation of Nonparametric Multiple Regression Measurement Error Models with Validation Data

DOI: 10.4236/ojs.2015.57080   PDF   HTML   XML   3,550 Downloads   4,132 Views   Citations

Abstract

In this article, we develop estimation approaches for nonparametric multiple regression measurement error models when both independent validation data on covariables and primary data on the response variable and surrogate covariables are available. An estimator which integrates Fourier series estimation and truncated series approximation methods is derived without any error model structure assumption between the true covariables and surrogate variables. Most importantly, our proposed methodology can be readily extended to the case that only some of covariates are measured with errors with the assistance of validation data. Under mild conditions, we derive the convergence rates of the proposed estimators. The finite-sample properties of the estimators are investigated through simulation studies.

Share and Cite:

Yin, Z. and Liu, F. (2015) Estimation of Nonparametric Multiple Regression Measurement Error Models with Validation Data. Open Journal of Statistics, 5, 808-819. doi: 10.4236/ojs.2015.57080.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Pepe, M.S. and Fleming, T.R. (1991) A General Nonparametric Method for Dealing with Errors in Missing or Surrogate Covaraite Data. Journal of the American Statistical Association, 86, 108-113.
http://dx.doi.org/10.1080/01621459.1991.10475009
[2] Pepe, M.S. (1992) Inference Using Surrogate Outcome Data and a Validation Sample. Biometrika, 79, 355-365.
http://dx.doi.org/10.1093/biomet/79.2.355
[3] Lee, L.F. and Sepanski, J. (1995) Estimation of Linear and Nonlinear Errors-in-Variables Models Using Validation Data. Journal of the American Statistical Association, 90, 130-140.
http://dx.doi.org/10.1080/01621459.1995.10476495
[4] Wang, Q. and Rao, J.N.K. (2002) Empirical Likelihood-Based Inference in Linear Errors-in-Covariables Models with Validation Data. Biometrika, 89, 345-358.
http://dx.doi.org/10.1093/biomet/89.2.345
[5] Zhang, Y. (2015) Estimation of Partially Linear Regression for Errors-in-Variables Models with Validation Data. Springer International Publishing, 322, 733-742.
http://dx.doi.org/10.1007/978-3-319-08991-1_76
[6] Xu, W. and Zhu, L. (2015) Nonparametric Check for Partial Linear Errors-in-Covariables Models with Validation Data. Annals of the Institute of Statistical Mathematics, 67, 793-815.
http://dx.doi.org/10.1007/s10463-014-0476-7
[7] Carroll, R.J. and Stefanski, L.A. (1990) Approximate Quasi-Likelihood Estimation in Models with Surrogate Predictors. Journal of the American Statistical Association, 85, 652-663.
http://dx.doi.org/10.1080/01621459.1990.10474925
[8] Carroll, R.J. and Wand, M.P. (1991) Semiparametric Estimation in Logistic Measurement Error Models. Journal of the Royal Statistical Society: Series B, 53, 573-585.
[9] Sepanski, J.H. and Lee, L.F. (1995) Semiparametric Estimation of Nonlinear Errors-in-Variables Models with Validation Study. Journal of Nonparametric Statistics, 4, 365-394.
http://dx.doi.org/10.1080/10485259508832627
[10] Stute, W., Xue, L. and Zhu, L. (2007) Empirical Likelihood Inference in Nonlinear Errors-in-Covariables Models with Validation Data. Journal of the American Statistical Association, 102, 332-346.
http://dx.doi.org/10.1198/016214506000000816
[11] Cook, J.R. and Stefanski, L.A. (1994) Simulation-Extrapolation Estimation in Parametric Measurement Error Models. Journal of the American Statistical Association, 89, 1314-1328.
http://dx.doi.org/10.1080/01621459.1994.10476871
[12] Carroll, R.J., Gail, M.H. and Lubin, J.H. (1993) Case-Control Studied with Errors in Covariables. Journal of the American Statistical Association, 88, 185-199.
[13] Lü, Y.-Z., Zhang, R.-Q. and Huang, Z.-S. (2013) Estimation of Semi-Varying Coefficient Model with Surrogate Data and Validation Sampling. Acta Mathematicae Applicatae Sinica, English Series, 29, 645-660.
http://dx.doi.org/10.1007/s10255-013-0241-3
[14] Xiao, Y. and Tian, Z. (2014) Dimension Reduction Estimation in Nonlinear Semiparametric Error-in-Response Models with Validation Data. Mathematica Applicata, 27, 730-737.
[15] Yu, S.H. and Wang, D.H. (2014) Empirical Likelihood for First-Order Autoregressive Error-in-Variable of Models with Validation Data. Communications in Statistics Theory Methods, 43, 1800-1823.
http://dx.doi.org/10.1080/03610926.2012.679763
[16] Stefanski, L.A. and Buzas, J.S. (1995) Instrumental Variable Estimation in Binary Regression Measurement Error Models. Journal of the American Statistical Association, 90, 541-550.
http://dx.doi.org/10.1080/01621459.1995.10476546
[17] Wang, Q. (2006) Nonparametric Regression Function Estimation with Surrogate Data and Validation sampling. Journal of Multivariate Analysis, 97, 1142-1161.
http://dx.doi.org/10.1016/j.jmva.2005.05.008
[18] Du, L., Zou, C. and Wang, Z. (2011) Nonparametric Regression Function Estimation for Error-in-Variable Models with Validation Data. Statistica Sinica, 21, 1093-1113.
http://dx.doi.org/10.5705/ss.2009.047
[19] Carroll, R.J., Ruppert, D., Stefanski, L.A. and Crainiceanu, C.M. (2006) Measurement Error in Nonlinear Models. Second Edition, Chapman and Hall CRC Press, Boca Raton.
http://dx.doi.org/10.1201/9781420010138
[20] Hall, P. and Horowitz, J.L. (2005) Nonparametric Methods for Inference in the Presence of Instrumental Variables. Annals of Statistics, 33, 2904-2929.
http://dx.doi.org/10.1214/009053605000000714
[21] Darolles, S., Florens, J.P. and Renault, E. (2006) Nonparametric Instrumental Regression. Working Paper, GREMAQ, University of Social Science, Toulouse.
[22] Newey, W.K. and Powell, J.L. (2003) Instrumental Variable Estimation of Nonparametric Models. Econometrica, 71, 1565-1578.
http://dx.doi.org/10.1111/1468-0262.00459
[23] Blundell, R., Chen, X. and Kristensen, D. (2007) Semi-Nonparametric IV Estimation of Shape-Invariant Engel Curves. Econometrica, 75, 1613-1669.
http://dx.doi.org/10.1111/j.1468-0262.2007.00808.x
[24] Newey, W.K. (1997) Convergence Rates and Asymptotic Normality for Series Estimators. Journal of Econometrics, 79, 147-168.
http://dx.doi.org/10.1016/S0304-4076(97)00011-0
[25] Schimek, M.G. (2012) Variance Estimation and Bandwidth Selection for Kernel Regression. John Wiley & Sons, Inc., New York, 71-107.
[26] Timan, A. (1963) Theory of Approximation of Functions of a Real Variable. McMillan, New York.

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.