Skip to main content
SpringerLink
Log in

Generalized varying coefficient partially linear measurement errors models

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We study generalized varying coefficient partially linear models when some linear covariates are error prone, but their ancillary variables are available. We first calibrate the error-prone covariates, then develop a quasi-likelihood-based estimation procedure. To select significant variables in the parametric part, we develop a penalized quasi-likelihood variable selection procedure, and the resulting penalized estimators are shown to be asymptotically normal and have the oracle property. Moreover, to select significant variables in the nonparametric component, we investigate asymptotic behavior of the semiparametric generalized likelihood ratio test. The limiting null distribution is shown to follow a Chi-square distribution, and a new Wilks phenomenon is unveiled in the context of error-prone semiparametric modeling. Simulation studies and a real data analysis are conducted to evaluate the performance of the proposed methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Akaike, H. (1973). Maximum likelihood identification of Gaussian autoregressive moving average models. Biometrika, 60, 255–265.

    Article  MathSciNet  MATH  Google Scholar 

  • Cai, Z., Fan, J., Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. Journal of the American Statistical Association, 95, 888–902.

  • Cambien, F., Warnet, J., Eschwege, E., Jacqueson, A., Richard, J., Rosselin, G. (1987). Body mass, blood pressure, glucose, and lipids. Does plasma insulin explain their relationships? Arteriosclerosis, Thrombosis, and Vascular Biology, 7, 197–202.

  • Carroll, R. J., Wang, Y. (2008). Nonparametric variance estimation in the analysis of microarray data: A measurement error approach. Biometrika, 95(2), 437–449.

  • Carroll, R. J., Fan, J., Gijbels, I., Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association, 92(438), 477–489.

  • Carroll, R. J., Ruppert, D., Stefanski, L. A., Crainiceanu, C. M. (2006). Nonlinear measurement error models, a modern perspective (2nd ed.). New York: Chapman and Hall.

  • Cook, J. R., Stefanski, L. A. (1994). Simulation-extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association, 89, 1314–1328.

  • Fan, J., Gijbels, I. (1996). Local polynomial modelling and its applications (Vol. 66). London: Chapman & Hall.

  • Fan, J., Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 11, 1031–1057.

  • Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96, 1348–1360.

  • Fan, J., Zhang, C. M., Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. The Annals of Statistics, 29, 153–193.

  • Foster, D., George, E. (1994). The risk inflation criterion for multiple regression. The Annals of Statistics, 22, 1947–1975.

  • Hall, P., Ma, Y. (2007). Semiparametric estimators of functional measurement error models with unknown error. Journal of the Royal Statistical Society, Series B Statistical Methodology, 69(3), 429–446.

  • Han, T. S., van Leer, E. M., Seidell, J. C., Lean, M. E. (1995). Waist circumference action levels in the identification of cardiovascular risk factors: prevalence study in a random sample. British Medical Journal (BMJ), 311(7017), 1401–1405.

  • Härdle, W., Liang, H., Gao, J. (2000). Partially linear models. Heidelberg: Physica-Verlag. Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). Journal of the Royal Statistical Society, Series B Statistical Methodology, 55, 757–796.

  • Hastie, T., Tibshirani, R. (1993). Varying-coefficient models (with discussion). Journal of the Royal Statistical Society, Series B Statistical Methodology, 55, 757–796.

  • Hunsberger, S. (1994). Semiparametric regression in likelihood-based models. Journal of the American Statistical Association, 89, 1354–1365.

    Article  MathSciNet  MATH  Google Scholar 

  • Hunsberger, S., Albert, P. S., Follmann, D. A., Suh, E. (2002). Parametric and semiparametric approaches to testing for seasonal trend in serial count data. Biostatistics, 3, 289–298.

  • Li, G., Xue, L., Lian, H. (2011). Semi-varying coefficient models with a diverging number of components. Journal of Multivariate Analysis, 102(7), 1166–1174.

  • Li, R., Liang, H. (2008). Variable selection in semiparametric regression modeling. The Annals of Statistics, 36, 261–286.

  • Liang, H., Li, R. (2009). Variable selection for partially linear models with measurement errors. Journal of the American Statistical Association, 104(485), 234–248.

  • Lin, X., Carroll, R. J. (2001). Semiparametric regression for clustered data using generalized estimating equations. Journal of the American Statistical Association, 96, 1045–1056.

  • Lobach, I., Carroll, R. J., Spinka, C., Gail, M., Chatterjee, N. (2008). Haplotype-based regression analysis and inference of case–control studies with unphased genotypes and measurement errors in environmental exposures. Biometrics, 64, 673–684.

  • Lobach, I., Fan, R., Carroll, R. J. (2010). Genotype-based association mapping of complex diseases: gene–environment interactions with multiple genetic markers and measurement error in environmental exposures. Genetic Epidemiology, 34, 792–802.

  • Ma, Y., Carroll, R. J. (2006). Locally efficient estimators for semiparametric models with measurement error. Journal of the American Statistical Association, 101(476), 1465–1474.

  • Ma, Y., Li, R. (2010). Variable selection in measurement error models. Bernoulli, 16(1), 274–300.

  • Ma, Y., Tsiatis, A. A. (2006). On closed form semiparametric estimators for measurement error models. Statistica Sinica, 16(1), 183–193.

  • Robinson, P. M. (1988). Root n-consistent semiparametric regression. Econometrica, 56, 931–954.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.

    Article  MathSciNet  MATH  Google Scholar 

  • Severini, T. A., Staniswalis, J. G. (1994). Quasi-likelihood estimation in semiparametric models. Journal of the American Statistical Association, 89, 501–511.

  • Silverman, B. W. (1986). Density estimation for statistics and data analysis, Vol. 26 of Monographs on statistics and applied probability. London: Chapman and Hall.

  • Sinha, S., Mallick, B. K., Kipnis, V., Carroll, R. J. (2010). Semiparametric Bayesian analysis of nutritional epidemiology data in the presence of measurement error. Biometrics, 66(2), 444–454.

  • Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B Statistical Methodology, 50, 413–436.

    MathSciNet  MATH  Google Scholar 

  • Stefanski, L. A. (1989). Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. Communications in Statistics. Theory and Methods, 18(12), 4335–4358.

    Article  MathSciNet  MATH  Google Scholar 

  • Stefanski, L. A., Carroll, R. J. (1987). Conditional scores and optimal scores for generalized linear measurement-error models. Biometrika, 74(4), 703–716.

  • Tsiatis, A. A., Ma, Y. (2004). Locally efficient semiparametric estimators for functional measurement error models. Biometrika, 91(4), 835–848.

  • Wang, H., Xia, Y. (2009). Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association, 104, 747–757.

  • Wang, L., Liu, X., Liang, H., Carroll, R. J. (2011). Estimation and variable selection for generalized additive partial linear models. The Annals of Statistics, 39, 1827–1851.

  • Wei, F., Huang, J., Li, H. (2011). Variable selection and estimation in high-dimensional varyingcoefficient models. Statistica Sinica, 21(4), 1515–1540.

  • Xia, Y., Zhang, W., Tong, H. (2004). Efficient estimation for semivarying-coefficient models. Biometrika, 91, 661–681.

  • Yi, G. Y., Ma, Y. Y., Carroll, R. J. (2012). A functional generalized method of moments approach for longitudinal studies with missing responses and covariate measurement error. Biometrika, 99(1), 151–165.

  • Yuan, M., Lin, Y. (2006). Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society, Series B Statistical Methodology, 68, 49–67.

  • Zhang, C. (2003). Calibrating the degrees of freedom for automatic data smoothing and effective curve checking. Journal of the American Statistical Association, 98(463), 609–628.

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, W., Lee, S.-Y., Song, X. (2002). Local polynomial fitting in semivarying coefficient model. Journal of Multivariate Analysis, 82, 166–188.

  • Zhou, Y., Liang, H. (2009). Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates. The Annals of Statistics, 37, 427–458.

Download references

Acknowledgments

The authors thank the associate editor, two referees for their constructive suggestions that helped us to improve the early manuscript. Zhang Jun’s research was supported by the National Natural Science Foundation of China (NSFC) Grant No. 11326179 (Tian yuan fund for Mathematics), and NSFC Grant No. 11401391, and the Project of Department of Education of Guangdong Province of China, Grant No. 2014KTSCX112. Feng Zhenghui’s research was supported by the NSFC Grant No. 11301434. Xu Peirong’s research was supported by the Natural Science Foundation of Jiangsu Province, China, Grant No. BK20140617. Liang Hua’s research was partially supported by NSF Grants DMS- 1440121 and DMS-1418042, and by Award Number 11228103, made by National Natural Science Foundation of China.

Author information

Authors and Affiliations

  1. Shen Zhen-Hong Kong Joint Research Centre for Applied Statistical Sciences, School of Mathematics and Statistics, Institute of Statistical Sciences, Shenzhen University, Shenzhen, 518060, China

    Jun Zhang

  2. School of Economics, and the Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, 361005, China

    Zhenghui Feng

  3. Department of Mathematics, Southeast University, Nanjing, 211189, China

    Peirong Xu

  4. Department of Statistics, George Washington University, Washington, DC, 20052, USA

    Hua Liang

Authors
  1. Jun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenghui Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peirong Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hua Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Zhang.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 143 KB)

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, J., Feng, Z., Xu, P. et al. Generalized varying coefficient partially linear measurement errors models. Ann Inst Stat Math 69, 97–120 (2017). https://doi.org/10.1007/s10463-015-0532-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-015-0532-y

Keywords

Search

Navigation