The Annals of Statistics

Regressions with Berkson errors in covariates—A nonparametric approach

Susanne M. Schennach

Full-text: Open access

Abstract

This paper establishes that so-called instrumental variables enable the identification and the estimation of a fully nonparametric regression model with Berkson-type measurement error in the regressors. An estimator is proposed and proven to be consistent. Its practical performance and feasibility are investigated via Monte Carlo simulations as well as through an epidemiological application investigating the effect of particulate air pollution on respiratory health. These examples illustrate that Berkson errors can clearly not be neglected in nonlinear regression models and that the proposed method represents an effective remedy.

Article information

Source
Ann. Statist., Volume 41, Number 3 (2013), 1642-1668.

Dates
First available in Project Euclid: 1 August 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1375362562

Digital Object Identifier
doi:10.1214/13-AOS1122

Mathematical Reviews number (MathSciNet)
MR3113824

Zentralblatt MATH identifier
1292.62061

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62H99: None of the above, but in this section

Keywords
Berkson measurement error errors in variables instrumental variables nonparametric inference nonparametric maximum likelihood

Citation

Schennach, Susanne M. Regressions with Berkson errors in covariates—A nonparametric approach. Ann. Statist. 41 (2013), no. 3, 1642--1668. doi:10.1214/13-AOS1122. https://projecteuclid.org/euclid.aos/1375362562


Export citation

References

  • Berkson, J. (1950). Are there two regressions? J. Amer. Statist. Assoc. 45 164–180.
  • Bhattacharya, R. N. and Rao, R. R. (2010). Normal Approximation and Asymptotic Expansions. SIAM, Philadelphia.
  • Carrasco, M., Florens, J. P. and Renault, E. (2005). Linear inverse problems and structural econometrics: Estimation based on spectral decomposition and regularization. In Handbook of Econometrics, Vol. 6. Elsevier, Amsterdam.
  • Carroll, R. J., Chen, X. and Hu, Y. (2010). Identification and estimation of nonlinear models using two samples with nonclassical measurement errors. J. Nonparametr. Stat. 22 379–399.
  • Carroll, R. J., Delaigle, A. and Hall, P. (2007). Non-parametric regression estimation from data contaminated by a mixture of Berkson and classical errors. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 859–878.
  • Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models. Chapman & Hall/CRC, Boca Raton, FL.
  • Delaigle, A., Hall, P. and Qiu, P. (2006). Nonparametric methods for solving the Berkson errors-in-variables problem. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 201–220.
  • D’Haultfoeuille, X. (2011). On the completeness condition in nonparametric instrumental problems. Econometric Theory 27 460–471.
  • Dockery, D. W., Pope, C. A., Xu, X. P. et al. (1993). An association between air-pollution and mortality in 6 United-States cities. New England Journal of Medicine 329 1753–1759.
  • Dunford, N. and Schwartz, J. T. (1971). Linear Operators. Wiley, New York.
  • Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables. Ann. Statist. 21 1900–1925.
  • Gallant, A. R. and Nychka, D. W. (1987). Semi-nonparametric maximum likelihood estimation. Econometrica 55 363–390.
  • Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures. Ann. Probab. 18 851–869.
  • Grenander, U. (1981). Abstract Inference. Wiley, New York.
  • Hausman, J. A., Newey, W. K. and Powell, J. L. (1995). Nonlinear errors in variables: Estimation of some Engel curves. J. Econometrics 65 205–233.
  • Hausman, J. A., Newey, W. K., Ichimura, H. and Powell, J. L. (1991). Identification and estimation of polynomial errors-in-variables models. J. Econometrics 50 273–295.
  • Hu, Y. and Schennach, S. M. (2008). Instrumental variable treatment of nonclassical measurement error models. Econometrica 76 195–216.
  • Huwang, L. and Huang, Y. H. S. (2000). On errors-in-variables in polynomial regression-Berkson case. Statist. Sinica 10 923–936.
  • Hyslop, D. R. and Imbens, G. W. (2001). Bias from classical and other forms of measurement error. J. Bus. Econom. Statist. 19 475–481.
  • Lewbel, A. (1996). Demand estimation with expenditure measurement errors on the left and right hand side. Rev. Econom. Statist. 78 718–725.
  • Li, T. (2002). Robust and consistent estimation of nonlinear errors-in-variables models. J. Econometrics 110 1–26.
  • Li, T. and Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. J. Multivariate Anal. 65 139–165.
  • Mahajan, A. (2006). Identification and estimation of regression models with misclassification. Econometrica 74 631–665.
  • Mallick, B., Hoffman, F. O. and Carroll, R. J. (2002). Semiparametric regression modeling with mixtures of Berkson and classical error, with application to fallout from the Nevada test site. Biometrics 58 13–20.
  • Nelder, J. A. and Mead, R. (1965). A simplex method for function minimization. Computer Journal 7 308–313.
  • Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. J. Econometrics 79 147–168.
  • Newey, W. (2001). Flexible simulated moment estimation of nonlinear errors-in-variables models. Rev. Econom. Statist. 83 616–627.
  • Newey, W. K. and Powell, J. L. (2003). Instrumental variable estimation of nonparametric models. Econometrica 71 1565–1578.
  • Pope, C. A., Thun, M. J., Namboodiri, M. M. et al. (1995). Particulate air-pollution as a predictor of mortality in a prospective-study of us adults. American Journal of Respiratory and Critical Care Medicine 151 669–674.
  • Samet, J. M., Dominici, F., Curriero, F. C. et al. (2000). Fine particulate air pollution and mortality in 20 US cities, 1987–1994. New England Journal of Medicine 343 1742–1749.
  • Schennach, S. M. (2004). Estimation of nonlinear models with measurement error. Econometrica 72 33–75.
  • Schennach, S. M. (2007). Instrumental variable estimation of nonlinear errors-in-variables models. Econometrica 75 201–239.
  • Schennach, S. M. (2013). Supplement to “Regressions with Berkson errors in covariates—A nonparametric approach.” DOI:10.1214/13-AOS1122SUPP.
  • Shen, X. (1997). On methods of sieves and penalization. Ann. Statist. 25 2555–2591.
  • Stram, D. O., Huberman, M. and Wu, A. H. (2002). Is residual confounding a reasonable explanation for the apparent protective effects of beta-carotene found in epidemiologic studies of lung cancer in smokers? Am. J. Epidemiol. 155 622–628.
  • van der Laan, M. J., Dudoit, S. and Keles, S. (2004). Asymptotic optimality of likelihood-based cross-validation. Stat. Appl. Genet. Mol. Biol. 3 Art. 4, 27 pp. (electronic).
  • Wang, L. (2004). Estimation of nonlinear models with Berkson measurement errors. Ann. Statist. 32 2559–2579.
  • Wang, L. (2007). A unified approach to estimation of nonlinear mixed effects and Berkson measurement error models. Canad. J. Statist. 35 233–248.

Supplemental materials

  • Supplementary material: Supplementary material to “Regressions with Berkson errors in covariates—A nonparametric approach”. The supplementary material provides (i) a proof of consistency of the proposed estimator, (ii) additional simulation results and (iii) various extensions of the method, including the weakening of some of full independence assumptions to conditional independence and handling the simultaneous presence of classical and Berkson errors.