zbMATH — the first resource for mathematics

Semiparametric quantile estimation for varying coefficient partially linear measurement errors models. (English) Zbl 1398.62101
Summary: 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.
62G08 Nonparametric regression and quantile regression
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
62P10 Applications of statistics to biology and medical sciences; meta analysis
Full Text: DOI Euclid
[1] Andrews, D. and Herzberg, A. (1985). Data. A Collection of Problems for Many Fields for Student and Research Worker. New York: Springer. · Zbl 0567.62002
[2] Bravo, F. (2013). Partially linear varying coefficient models with missing at random responses. Annals of the Institute of Statistical Mathematics65, 721-762. · Zbl 1273.62081
[3] Cai, Z., Fan, J. and Li, R. (2000). Efficient estimation and inferences for varying-coefficient models. Journal of the American Statistical Association95, 888-902. · Zbl 0999.62052
[4] Cai, Z., Naik, P. A. and Tsai, C.-L. (2000). Denoised least squares estimators: An application to estimating advertising effectiveness. Statistica Sinica10, 1231-1241. · Zbl 0960.62134
[5] Cai, Z. and Xiao, Z. (2012). Semiparametric quantile regression estimation in dynamic models with partially varying coefficients. Journal of Econometrics167, 413-425. · Zbl 1441.62623
[6] Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. Journal of the American Statistical Association92, 477-489. · Zbl 0890.62053
[7] 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. · Zbl 1119.62063
[8] Cui, H., He, X. and Zhu, L. (2002). On regression estimators with de-noised variables. Statistica Sinica12, 1191-1205. · Zbl 1004.62038
[9] Cui, H. and Hu, T. (2011). On nonlinear regression estimator with denoised variables. Computational Statistics & Data Analysis55, 1137-1149. · Zbl 1284.62402
[10] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications, Vol. 66. London: Chapman & Hall. · Zbl 0873.62037
[11] Fan, J. and Zhang, W. (2008). Statistical methods with varying coefficient models. Statistics and its Interface1, 179-195. · Zbl 1230.62031
[12] Fan, Y. and Zhu, L. (2013). Estimation of general semi-parametric quantile regression. Journal of Statistical Planning and Inference143, 896-910. · Zbl 1428.62154
[13] Feng, L., Zou, C. and Wang, Z. (2012). Local Walsh-average regression. Journal of Multivariate Analysis106, 36-48. · Zbl 1297.62087
[14] Fuller, W. A. (1987). Measurement Error Models. New York: Wiley. · Zbl 0800.62413
[15] Gu, J. and Liang, Z. (2014). Testing cointegration relationship in a semiparametric varying coefficient model. Journal of Econometrics178, 57-70. · Zbl 1293.62099
[16] 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 Analysis65, 56-67.
[17] Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models. Heidelberg: Physica-Verlag.
[18] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). Journal of the Royal Statistical Society, Series B55, 757-796. · Zbl 0796.62060
[19] He, X. and Liang, H. (2000). Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Statistica Sinica10, 129-140. · Zbl 0970.62043
[20] Heckman, N. E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B48, 244-248. · Zbl 0623.62030
[21] Hu, Y., Gramacy, R. B. and Lian, H. (2013). Bayesian quantile regression for single-index models. Statistics and Computing23, 437-454. · Zbl 1325.62089
[22] Jiang, R., Zhou, Z.-G., Qian, W.-M. and Chen, Y. (2013). Two step composite quantile regression for single-index models. Computational Statistics & Data Analysis64, 180-191.
[23] Jiang, X., Jiang, J. and Song, X. (2012). Oracle model selection for nonlinear models based on weighted composite quantile regression. Statistica Sinica22, 1479-1506. · Zbl 1253.62025
[24] 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 Methodology72, 49-69.
[25] 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 Statistics39, 305-332. · Zbl 1209.62074
[26] Knight, K. (1998). Limiting distributions for \(L_{1}\) regression estimators under general conditions. The Annals of Statistics26, 755-770. · Zbl 0929.62021
[27] Koenker, R. (2005). Quantile Regression. Cambridge: Cambridge Univ. Pres. · Zbl 1111.62037
[28] Li, G., Feng, S. and Peng, H. (2011). A profile-type smoothed score function for a varying coefficient partially linear model. Journal of Multivariate Analysis102 372-385. · Zbl 1327.62263
[29] Li, G., Xue, L. and Lian, H. (2011). Semi-varying coefficient models with a diverging number of components. Journal of Multivariate Analysis102, 1166-1174. · Zbl 1216.62060
[30] Li, Q., Ouyang, D. and Racine, J. S. (2013). Categorical semiparametric varying-coefficient models. Journal of Applied Econometrics28, 551-579.
[31] Li, R. and Liang, H. (2008). Variable selection in semiparametric regression modeling. The Annals of Statistics36, 261-286. · Zbl 1132.62027
[32] Liang, H. (2008). Related Topics in Partially Linear Models. Saarbrucken, Germany: VDM Verlag.
[33] Mack, Y. P. and Silverman, B. W. (1982). Weak and strong uniform consistency of kernel regression estimates. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete61, 405-415. · Zbl 0495.62046
[34] Pollard, D. (1984). Convergence of Stochastic Processes. Springer Series in Statistics. New York: Springer.
[35] Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory7, 186-199.
[36] Robinson, P. M. (1988). Root \(n\)-consistent semiparametric regression. Econometrica56, 931-954. · Zbl 0647.62100
[37] Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley. · Zbl 0538.62002
[38] Shang, S., Zou, C. and Wang, Z. (2012). Local Walsh-average regression for semiparametric varying-coefficient models. Statistics & Probability Letters82, 1815-1822. · Zbl 1348.62147
[39] Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Monographs on Statistics and Applied Probability26. London: Chapman and Hall. · Zbl 0617.62042
[40] Speckman, P. E. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B50, 413-436. · Zbl 0671.62045
[41] 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 Inference143, 1049-1063. · Zbl 1428.62166
[42] 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. · Zbl 0862.60002
[43] Wang, H. and Xia, Y. (2009). Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association104, 747-757. · Zbl 1388.62213
[44] Wang, H. J., Stefanski, L. A. and Zhu, Z. (2012). Corrected-loss estimation for quantile regression with covariate measurement errors. Biometrika99, 405-421. · Zbl 1239.62047
[45] Wang, H. J., Zhu, Z. and Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. The Annals of Statistics37, 3841-3866. · Zbl 1191.62077
[46] Wang, L., Kai, B. and Li, R. (2009). Local rank inference for varying coefficient models. Journal of the American Statistical Association104, 1631-1645. · Zbl 1205.62092
[47] Wei, F., Huang, J. and Li, H. (2011). Variable selection and estimation in high-dimensional varying-coefficient models. Statistica Sinica21, 1515-1540. · Zbl 1225.62056
[48] Wei, Y. and Carroll, R. J. (2009). Quantile regression with measurement error. Journal of the American Statistical Association104, 1129-1143. · Zbl 1388.62210
[49] 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 Methodology64, 363-410. · Zbl 1091.62028
[50] Xia, Y., Zhang, W. and Tong, H. (2004). Efficient estimation for semivarying-coefficient models. Biometrika91, 661-681. · Zbl 1108.62019
[51] 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 Statistics7, 102-125. · Zbl 1454.62425
[52] Zhang, W., Lee, S.-Y. and Song, X. (2002). Local polynomial fitting in semivarying coefficient model. Journal of Multivariate Analysis82, 166-188. · Zbl 0995.62038
[53] Zhou, Y. and Liang, H. (2009). Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates. The Annals of Statistics37, 427-458. · Zbl 1156.62036
[54] Zhu, H., Li, R. and Kong, L. (2012). Multivariate varying coefficient model for functional responses. The Annals of Statistics40, 2634-2666. · Zbl 1373.62169
[55] Zou, H. and Yuan, M. (2008). Composite quantile regression and the oracle model selection theory. The Annals of Statistics36, 1108-1126. · Zbl 1360.62394
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.