Electronic Journal of Statistics

Estimation and inference of error-prone covariate effect in the presence of confounding variables

Jianxuan Liu, Yanyuan Ma, Liping Zhu, and Raymond J. Carroll

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We introduce a general single index semiparametric measurement error model for the case that the main covariate of interest is measured with error and modeled parametrically, and where there are many other variables also important to the modeling. We propose a semiparametric bias-correction approach to estimate the effect of the covariate of interest. The resultant estimators are shown to be root-$n$ consistent, asymptotically normal and locally efficient. Comprehensive simulations and an analysis of an empirical data set are performed to demonstrate the finite sample performance and the bias reduction of the locally efficient estimators.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 480-501.

Received: March 2015
First available in Project Euclid: 2 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62H15: Hypothesis testing
Secondary: 62F12: Asymptotic properties of estimators

Confounding effect measurement error primary effect semiparametric efficiency single index model

Creative Commons Attribution 4.0 International License.


Liu, Jianxuan; Ma, Yanyuan; Zhu, Liping; Carroll, Raymond J. Estimation and inference of error-prone covariate effect in the presence of confounding variables. Electron. J. Statist. 11 (2017), no. 1, 480--501. doi:10.1214/17-EJS1242. https://projecteuclid.org/euclid.ejs/1488423805

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  • Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993)., Efficient and Adaptive Estimation for Semiparametric Models. Baltimore, MD: Johns Hopkins University Press.
  • Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models., Journal of the American Statistical Association, 92, 477–489.
  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density., Journal of the American Statistical Association, 83, 1184–1186.
  • Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. (2006)., Measurement Error in Nonlinear Models: A Modern Perspective, Second edition. Boca Raton, CRC Press.
  • Cui, X., Härdle, W. and Zhu, L. X. (2011). The EFM approach for single-index models’., Annals of Statistics, 39, 1658–1688.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems., Annals of Statistics, 19, 1257–1272.
  • Fan, J. and Gijbels, I. (1996)., Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Härdle, W., Liang, H. & Gao, J. (2000)., Partially Linear Models. Heidelberg, Physica-Verlag.
  • Heckman, N. E. (1986). Spline smoothing in a partly linear model., Journal of the Royal Statistical Society, Series B, 48, 244–8.
  • Hubert, H. B., Feinleib, M., McNamara, P. M. and Castelli, W. P. (1983). Obesity as an independent risk factor for cardiovascular disease: a 26-year follow-up of participants in the Framingham Heart Study., Circulation, 67, 968–977.
  • Li, L., Zhu, L. P. and Zhu, L. X. (2011). Inference on the primary parameter of interest with the aid of dimension reduction estimation., Journal of the Royal Statistical Society, Series B, 73, 59–80.
  • Liang, H., Härdle, W., and Carroll, R. J. (1999). Estimation in a semiparametric partially linear errors-in-variables model., Annals of Statistics, 27, 1519–1535.
  • Ma, Y. and Carroll, R. J. (2006). Locally efficient estimators for semiparametric models with measurement error., Journal of the American Statistical Association, 101, 1465–1474.
  • Ma, Y., Chiou, J.-M. and Wang, N. (2006). Efficient semiparametric estimator for heteroscedastic partially-linear models., Biometrika, 93, 75–84.
  • Ma, Y. and Tsiatis, A. A. (2006). Closed form semiparametric estimators for measurement error models., Statistica Sinica, 16, 183–193.
  • Ma, Y. and Zhu, L. (2013). Doubly robust and efficient estimators for heteroscedastic partially linear single-index model allowing high dimensional covariates., Journal of the Royal Statistical Society, Series B, 75, 305–322.
  • Metropolitan Life Insurance Company (1959). New weight standards for men and women., Statistical Bulletin of the Metropolitan Life Insurance Company, 40, 1.
  • Newey, W. K. (1990). Semiparametric efficiency bounds., Journal of Applied Econometrics, 5, 99–135.
  • Stefanski, L. A. and Carroll, R. J. (1987). Conditional scores and optimal scores for generalized linear measurement-error models., Biometrika, 74, 703–716.
  • Tsiatis, A. A. (2006)., Semiparametric Theory and Missing Data. Springer, New York.
  • Tsiatis, A. A., and Ma, Y. (2004). Locally efficient semiparametric estimators for functional measurement error model., Biometrika, 91, 835–848.
  • Van Keilegom, I. and Carroll, R. J. (2007) Backfitting versus profiling in general criterion functions., Statistica Sinica, 17, 797–816.
  • Yu, Y. and Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models., Journal of the American Statistical Association, 97, 1042–1054.