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Quantile regression in partially linear varying coefficient models. (English) Zbl 1191.62077
Summary: Semiparametric models are often considered for analyzing longitudinal data for a good balance between flexibility and parsimony. We study a class of marginal partially linear quantile models with possibly varying coefficients. The functional coefficients are estimated by basis function approximations. The estimation procedure is easy to implement, and it requires no specification of the error distributions. The asymptotic properties of the proposed estimators are established for the varying coefficients as well as for the constant coefficients. We develop rank score tests for hypotheses on the coefficients, including the hypotheses of the constancy of a subset of the varying coefficients. Hypothesis testing of this type is theoretically challenging, as the dimensions of the parameter spaces under both the null and the alternative hypotheses are growing with the sample size. We assess the finite sample performance of the proposed method by Monte Carlo simulation studies, and demonstrate its value by the analysis of an AIDS data set, where the modeling of quantiles provides more comprehensive information than the usual least squares approach.

62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)
65C05 Monte Carlo methods
62P10 Applications of statistics to biology and medical sciences; meta analysis
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[1] Ahmad, I., Leelahanon, S. and Li, Q. (2005). Efficient estimation of a semiparametric partially linear varying coefficient model. Ann. Statist. 33 258-283. · Zbl 1064.62043
[2] Cai, Z. and Xu, X. (2008). Nonparametric quantile estimations for dynamic smooth coefficient models. J. Amer. Statist. Assoc. 103 1595-1608. · Zbl 1286.62029
[3] Chiang, C. T., Rice, J. A. and Wu, C. O. (2001). Smoothing spline estimation for varying-coefficient models with repeatedly measured dependent variables. J. Amer. Statist. Assoc. 96 605-619. JSTOR: · Zbl 1018.62034
[4] Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 1031-1057. · Zbl 1098.62077
[5] Fan, J., Huang, T. and Li, R. (2007). Analysis of longitudinal data with semiparametric estimation of covariance function. J. Amer. Statist. Assoc. 102 632-641. · Zbl 1172.62323
[6] Gutenbrunner, C., Jurêcková, J., Koenker, R. and Portnoy, S. (1993). Tests of linear hypotheses based on regression rank scores. J. Nonparametr. Stat. 2 307-333. · Zbl 1360.62216
[7] Hall, P. and Sheather, S. J. (1988). On the distribution of a studentized quantile. J. Roy. Statist. Soc. Ser. B 50 381-391. JSTOR: · Zbl 0674.62034
[8] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models. J. Roy. Statist. Soc. Ser. B 55 757-796. JSTOR: · Zbl 0796.62060
[9] He, X. and Shao, Q. M. (2000). On parameters of increasing dimensions. J. Multivariate Anal. 73 120-135. · Zbl 0948.62013
[10] He, X., Zhu, Z. Y. and Fung, W. K. (2002). Estimation in a semiparametric model for longitudinal data with unspecified dependence structure. Biometrika 89 579-590. JSTOR: · Zbl 1036.62035
[11] Hendricks, W. and Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity. J. Amer. Statist. Assoc. 87 58-68.
[12] Honda, T. (2004). Quantile regression in varying coefficient models. J. Statist. Plann. Inference 121 113-125. · Zbl 1038.62041
[13] Huang, J. Z., Wu, C. O. and Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika 89 111-128. JSTOR: · Zbl 0998.62024
[14] Kaslow, R. A., Ostrow, D. G., Detels, R., Phair, J. P., Polk, B. F. and Rinaldo, C. R. (1987). The multicenter AIDS cohort study: Rationale, organization and selected characteristics of the participants. American Journal of Epidemiology 126 310-318.
[15] Kim, M. (2007). Quantile regression with varying coefficients. Ann. Statist. 35 92-108. · Zbl 1114.62051
[16] Koenker, R. (1994). Confidence intervals for regression quantiles. In Asymptotic Statistics: Proceedings of the 5th Prague Symposium on Asymptotic Statistics (P. Mandl and M. Husková, eds.) 349-359. Physica, Heidelberg.
[17] Koenker, R. (2004). Quantile regression for longitudinal data. J. Multivariate Anal. 91 74-89. · Zbl 1051.62059
[18] Lipsitz, S. R., Fitzmaurice, G. M., Molenberghs, G. and Zhao, L. P. (1997). Quantile regression methods for longitudinal data with drop-outs: Application to CD4 cell counts of patients infected with the human immunodeficiency virus. J. Roy. Statist. Soc. Ser. C 46 463-476. · Zbl 0908.62114
[19] Mu, Y. and Wei, Y. (2009). A dynamic quantile regression transformation model for longitudinal data. Statist. Sinica 19 1137-1153. · Zbl 1166.62020
[20] Portoy, S. (1985). Asymptotic behavior of M -estimators of p regression parameters when p 2 / n is large; II. Normal approximation. Ann. Statist. 13 1403-1417. · Zbl 0601.62026
[21] Qu, A. and Li, R. (2006). Quadratic inference functions for varying coefficient models with longitudinal data. Biometrics 62 379-391. · Zbl 1097.62037
[22] Schumaker, L. L. (1981). Spline Functions: Basic Theory . Wiley, New York. · Zbl 0449.41004
[23] Sun, Y. and Wu, H. (2005). Semiparametric time-varying coefficients regression model for longitudinal data. Scand. J. Statist. 32 21-47. · Zbl 1091.62088
[24] Wang, H. and He, X. (2007). Detecting differential expressions in GeneChip microarray studies: A quantile approach. J. Amer. Statist. Assoc. 102 104-112. · Zbl 1284.62439
[25] Wei, Y. and He, X. (2006). Conditional growth charts (with discussion). Ann. Statist. 34 2069-2097. · Zbl 1106.62049
[26] Yu, K. and Jones, M. C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228-237. JSTOR: · Zbl 0906.62038
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