Brazilian Journal of Probability and Statistics

Semiparametric quantile estimation for varying coefficient partially linear measurement errors models

Jun Zhang, Yan Zhou, Xia Cui, and Wangli Xu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study varying coefficient partially linear models when some linear covariates are error-prone, but their ancillary variables are available. After calibrating the error-prone covariates, we study quantile regression estimates for parametric coefficients and nonparametric varying coefficient functions, and we develop a semiparametric composite quantile estimation procedure. Asymptotic properties of the proposed estimators are established, and the estimators achieve their best convergence rate with proper bandwidth conditions. Simulation studies are conducted to evaluate the performance of the proposed method, and a real data set is analyzed as an illustration.

Article information

Source
Braz. J. Probab. Stat., Volume 32, Number 3 (2018), 616-656.

Dates
Received: December 2016
Accepted: February 2017
First available in Project Euclid: 8 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1528444875

Digital Object Identifier
doi:10.1214/17-BJPS357

Mathematical Reviews number (MathSciNet)
MR3812385

Zentralblatt MATH identifier
06930042

Keywords
Ancillary variables quantile regression composite quantile regression errors-in-variables varying coefficient models

Citation

Zhang, Jun; Zhou, Yan; Cui, Xia; Xu, Wangli. Semiparametric quantile estimation for varying coefficient partially linear measurement errors models. Braz. J. Probab. Stat. 32 (2018), no. 3, 616--656. doi:10.1214/17-BJPS357. https://projecteuclid.org/euclid.bjps/1528444875


Export citation

References

  • Andrews, D. and Herzberg, A. (1985). Data. A Collection of Problems for Many Fields for Student and Research Worker. New York: Springer.
  • Bravo, F. (2013). Partially linear varying coefficient models with missing at random responses. Annals of the Institute of Statistical Mathematics 65, 721–762.
  • Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. Journal of the American Statistical Association 95, 888–902.
  • Cai, Z., Naik, P. A. and Tsai, C.-L. (2000). Denoised least squares estimators: An application to estimating advertising effectiveness. Statistica Sinica 10, 1231–1241.
  • Cai, Z. and Xiao, Z. (2012). Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. Journal of Econometrics 167, 413–425.
  • 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., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Nonlinear Measurement Error Models, a Modern Perspective, 2nd ed. New York: Chapman and Hall.
  • Cui, H., He, X. and Zhu, L. (2002). On regression estimators with de-noised variables. Statistica Sinica 12, 1191–1205.
  • Cui, H. and Hu, T. (2011). On nonlinear regression estimator with denoised variables. Computational Statistics & Data Analysis 55, 1137–1149.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications, Vol. 66. London: Chapman & Hall.
  • Fan, J. and Zhang, W. (2008). Statistical methods with varying coefficient models. Statistics and its Interface 1, 179–195.
  • Fan, Y. and Zhu, L. (2013). Estimation of general semi-parametric quantile regression. Journal of Statistical Planning and Inference 143, 896–910.
  • Feng, L., Zou, C. and Wang, Z. (2012). Local Walsh-average regression. Journal of Multivariate Analysis 106, 36–48.
  • Fuller, W. A. (1987). Measurement Error Models. New York: Wiley.
  • Gu, J. and Liang, Z. (2014). Testing cointegration relationship in a semiparametric varying coefficient model. Journal of Econometrics 178, 57–70.
  • Guo, J., Tang, M., Tian, M. and Zhu, K. (2013). Variable selection in high-dimensional partially linear additive models for composite quantile regression. Computational Statistics & Data Analysis 65, 56–67.
  • Härdle, W., Liang, H. and 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 55, 757–796.
  • He, X. and Liang, H. (2000). Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Statistica Sinica 10, 129–140.
  • Heckman, N. E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B 48, 244–248.
  • Hu, Y., Gramacy, R. B. and Lian, H. (2013). Bayesian quantile regression for single-index models. Statistics and Computing 23, 437–454.
  • Jiang, R., Zhou, Z.-G., Qian, W.-M. and Chen, Y. (2013). Two step composite quantile regression for single-index models. Computational Statistics & Data Analysis 64, 180–191.
  • Jiang, X., Jiang, J. and Song, X. (2012). Oracle model selection for nonlinear models based on weighted composite quantile regression. Statistica Sinica 22, 1479–1506.
  • Kai, B., Li, R. and Zou, H. (2010). Local composite quantile regression smoothing: An efficient and safe alternative to local polynomial regression. Journal of the Royal Statistical Society, Series B, Statistical Methodology 72, 49–69.
  • Kai, B., Li, R. and Zou, H. (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. The Annals of Statistics 39, 305–332.
  • Knight, K. (1998). Limiting distributions for $L_{1}$ regression estimators under general conditions. The Annals of Statistics 26, 755–770.
  • Koenker, R. (2005). Quantile Regression. Cambridge: Cambridge Univ. Pres.
  • Li, G., Feng, S. and Peng, H. (2011). A profile-type smoothed score function for a varying coefficient partially linear model. Journal of Multivariate Analysis 102 372–385.
  • Li, G., Xue, L. and Lian, H. (2011). Semi-varying coefficient models with a diverging number of components. Journal of Multivariate Analysis 102, 1166–1174.
  • Li, Q., Ouyang, D. and Racine, J. S. (2013). Categorical semiparametric varying-coefficient models. Journal of Applied Econometrics 28, 551–579.
  • Li, R. and Liang, H. (2008). Variable selection in semiparametric regression modeling. The Annals of Statistics 36, 261–286.
  • Liang, H. (2008). Related Topics in Partially Linear Models. Saarbrucken, Germany: VDM Verlag.
  • Mack, Y. P. and Silverman, B. W. (1982). Weak and strong uniform consistency of kernel regression estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 61, 405–415.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. New York: Springer.
  • Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–199.
  • Robinson, P. M. (1988). Root $n$-consistent semiparametric regression. Econometrica 56, 931–954.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
  • Shang, S., Zou, C. and Wang, Z. (2012). Local Walsh-average regression for semiparametric varying-coefficient models. Statistics & Probability Letters 82, 1815–1822.
  • Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability 26. London: Chapman and Hall.
  • Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B 50, 413–436.
  • Sun, J., Gai, Y. and Lin, L. (2013). Weighted local linear composite quantile estimation for the case of general error distributions. Journal of Statistical Planning and Inference 143, 1049–1063.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. New York: Springer. With applications to statistics.
  • Wang, H. and Xia, Y. (2009). Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association 104, 747–757.
  • Wang, H. J., Stefanski, L. A. and Zhu, Z. (2012). Corrected-loss estimation for quantile regression with covariate measurement errors. Biometrika 99, 405–421.
  • Wang, H. J., Zhu, Z. and Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. The Annals of Statistics 37, 3841–3866.
  • Wang, L., Kai, B. and Li, R. (2009). Local rank inference for varying coefficient models. Journal of the American Statistical Association 104, 1631–1645.
  • Wei, F., Huang, J. and Li, H. (2011). Variable selection and estimation in high-dimensional varying-coefficient models. Statistica Sinica 21, 1515–1540.
  • Wei, Y. and Carroll, R. J. (2009). Quantile regression with measurement error. Journal of the American Statistical Association 104, 1129–1143.
  • Xia, Y., Tong, H., Li, W. K. and Zhu, L.-X. (2002). An adaptive estimation of dimension reduction space. Journal of the Royal Statistical Society, Series B, Statistical Methodology 64, 363–410.
  • Xia, Y., Zhang, W. and Tong, H. (2004). Efficient estimation for semivarying-coefficient models. Biometrika 91, 661–681.
  • Yuan, Y., Zhu, H., Styner, M., Gilmore, J. H. and Marron, J. S. (2013). Varying coefficient model for modeling diffusion tensors along white matter tracts. Annals of Applied Statistics 7, 102–125.
  • Zhang, W., Lee, S.-Y. and Song, X. (2002). Local polynomial fitting in semivarying coefficient model. Journal of Multivariate Analysis 82, 166–188.
  • Zhou, Y. and Liang, H. (2009). Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates. The Annals of Statistics 37, 427–458.
  • Zhu, H., Li, R. and Kong, L. (2012). Multivariate varying coefficient model for functional responses. The Annals of Statistics 40, 2634–2666.
  • Zou, H. and Yuan, M. (2008). Composite quantile regression and the oracle model selection theory. The Annals of Statistics 36, 1108–1126.