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Monte Carlo methods for nonparametric regression with heteroscedastic measurement error

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  • Julie McIntyre
  • Brent A. Johnson
  • Stephen M. Rappaport

Abstract

Nonparametric regression is a fundamental problem in statistics but challenging when the independent variable is measured with error. Among the first approaches was an extension of deconvoluting kernel density estimators for homescedastic measurement error. The main contribution of this article is to propose a new simulation†based nonparametric regression estimator for the heteroscedastic measurement error case. Similar to some earlier proposals, our estimator is built on principles underlying deconvoluting kernel density estimators. However, the proposed estimation procedure uses Monte Carlo methods for estimating nonlinear functions of a normal mean, which is different than any previous estimator. We show that the estimator has desirable operating characteristics in both large and small samples and apply the method to a study of benzene exposure in Chinese factory workers.

Suggested Citation

  • Julie McIntyre & Brent A. Johnson & Stephen M. Rappaport, 2018. "Monte Carlo methods for nonparametric regression with heteroscedastic measurement error," Biometrics, The International Biometric Society, vol. 74(2), pages 498-505, June.
    • Handle: RePEc:bla:biomet:v:74:y:2018:i:2:p:498-505
    DOI: 10.1111/biom.12765
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    References listed on IDEAS

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    1. Delaigle, Aurore & Meister, Alexander, 2007. "Nonparametric Regression Estimation in the Heteroscedastic Errors-in-Variables Problem," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1416-1426, December.
    2. Julie McIntyre & Leonard Stefanski, 2011. "Density Estimation with Replicate Heteroscedastic Measurements," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 63(1), pages 81-99, February.
    3. Raymond J. Carroll & David Ruppert & Ciprian M. Crainiceanu & Tor D. Tosteson & Margaret R. Karagas, 2004. "Nonlinear and Nonparametric Regression and Instrumental Variables," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 736-750, January.
    4. Raymond J. Carroll & Peter Hall, 2004. "Low order approximations in deconvolution and regression with errors in variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 31-46, February.
    5. A. Delaigle & I. Gijbels, 2004. "Bootstrap bandwidth selection in kernel density estimation from a contaminated sample," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(1), pages 19-47, March.
    6. Delaigle, A. & Gijbels, I., 2004. "Practical bandwidth selection in deconvolution kernel density estimation," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 249-267, March.
    7. Aurore Delaigle & Peter Hall & Farshid Jamshidi, 2015. "Confidence bands in non-parametric errors-in-variables regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(1), pages 149-169, January.
    8. John Staudenmayer & David Ruppert, 2004. "Local polynomial regression and simulation–extrapolation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 17-30, February.
    9. Delaigle, Aurore & Fan, Jianqing & Carroll, Raymond J., 2009. "A Design-Adaptive Local Polynomial Estimator for the Errors-in-Variables Problem," Journal of the American Statistical Association, American Statistical Association, vol. 104(485), pages 348-359.
    10. Delaigle, Aurore & Hall, Peter, 2008. "Using SIMEX for Smoothing-Parameter Choice in Errors-in-Variables Problems," Journal of the American Statistical Association, American Statistical Association, vol. 103, pages 280-287, March.
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