In this paper, we investigate the quantile varying coefficient model for longitudinal data, where the unknown nonparametric functions are approximated by polynomial splines and the estimators are obtained by minimizing the quadratic inference function. The theoretical properties of the resulting estimators are established, and they achieve the optimal convergence rate for the nonparametric functions. Since the objective function is non-smooth, an estimation procedure is proposed that uses induced smoothing and we prove that the smoothed estimator is asymptotically equivalent to the original estimator. Moreover, we propose a variable selection procedure based on the regularization method, which can simultaneously estimate and select important nonparametric components and has the asymptotic oracle property. Extensive simulations and a real data analysis show the usefulness of the proposed method.
This is a preview of subscription content, log in to check access.
Buy single article
Instant unlimited access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Andriyana, Y., Gijbels, I., Verhasselt, A. (2014). P-splines quantile regression estimation in varying coefficient models. Test, 23(1), 153–194.
Bondell, H. D., Reich, B. J., Wang, H. (2010). Noncrossing quantile regression curve estimation. Biometrika, 97(4), 825–838.
Brown, B., Wang, Y.-G. (2005). Standard errors and covariance matrices for smoothed rank estimators. Biometrika, 92(1), 149–158.
Cai, Z., Xu, X. (2008). Nonparametric quantile estimations for dynamic smooth coefficient models. Journal of the American Statistical Association, 103(484), 371–383.
Chiang, C.-T., Rice, J. A., Wu, C. O. (2001). Smoothing spline estimation for varying coefficient models with repeatedly measured dependent variables. Journal of the American Statistical Association, 96(454), 605–619.
De Boor, C. (2001). A practical guide to splines (revised ed., Vol. 27)., Applied mathematical sciences New York: Springer.
Fan, J., Huang, T., Li, R. (2007). Analysis of longitudinal data with semiparametric estimation of covariance function. Journal of the American Statistical Association, 102(478), 632–641.
Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96(456), 1348–1360.
Fan, J., Zhang, W. (1999). Statistical estimation in varying coefficient models. Annals of Statistics, 27, 1491–1518.
Fan, J., Zhang, W. (2000). Simultaneous confidence bands and hypothesis testing in varying-coefficient models. Scandinavian Journal of Statistics, 27(4), 715–731.
Fu, L., Wang, Y.-G. (2012). Quantile regression for longitudinal data with a working correlation model. Computational Statistics & Data Analysis, 56(8), 2526–2538.
Greene, W. H. (2011). Econometric Analysis. New York, USA: Pearson.
Hall, P., Sheather, S. J. (1988). On the distribution of a studentized quantile, Journal of the Royal Statistical Society. Series B (Methodological), 50(3), 381–391.
Hastie, T., Tibshirani, R. (1993). Varying-coefficient models. Journal of the Royal Statistical Society Series B (Methodological), 55(4), 757–796.
He, X., Shi, P. (1996). Bivariate tensor-product B-splines in a partly linear model. Journal of Multivariate Analysis, 58(2), 162–181.
Hendricks, W., Koenker, R. (1992). Hierarchical spline models for conditional quantiles and the demand for electricity. Journal of the American Statistical Association, 87(417), 58–68.
Huang, J. Z., Wu, C. O., Zhou, L. (2002). Varying-coefficient models and basis function approximations for the analysis of repeated measurements. Biometrika, 89(1), 111–128.
Huang, J. Z., Zhang, L., Zhou, L. (2007). Efficient estimation in marginal partially linear models for longitudinal/clustered data using splines. Scandinavian Journal of Statistics, 34(3), 451–477.
Jung, S.-H. (1996). Quasi-likelihood for median regression models. Journal of the American Statistical Association, 91(433), 251–257.
Kaslow, R. A., Ostrow, D. G., Detels, R., Phair, J. P., Polk, B. F., Rinaldo, C. R., et al. (1987). The multicenter aids cohort study: rationale, organization, and selected characteristics of the participants. American Journal of Epidemiology, 126(2), 310–318.
Kim, M.-O. (2007). Quantile regression with varying coefficients. The Annals of Statistics, 35(1), 92–108.
Koenker, R. (2005). Quantile regression. Cambridge, UK: Cambridge University Press.
Leng, C., Zhang, W. (2014). Smoothing combined estimating equations in quantile regression for longitudinal data. Statistics and Computing, 24(1), 123–136.
Li, G., Lai, P., Lian, H. (2014). Variable selection and estimation for partially linear single-index models with longitudinal data, Statistics and Computing (in press).
Lian, H., Liang, H., Wang, L. (2014). Generalized additive partial linear models for clustered data with diverging number of covariates using gee. Statistica Sinica, 24(1), 173–196.
Liang, K.-Y., Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73(1), 13–22.
Lin, H., Song, P. X.-K., Zhou, Q. M. (2007). Varying-coefficient marginal models and applications in longitudinal data analysis. Sankhyā: The Indian Journal of Statistics, 69(3), 581–614.
Ma, S., Liang, H., Tsai, C.-L. (2014). Partially linear single index models for repeated measurements. Journal of Multivariate Analysis, 130, 354–375.
Noh, H., Chung, K., Van Keilegom, I., et al. (2012). Variable selection of varying coefficient models in quantile regression. Electronic Journal of Statistics, 6, 1220–1238.
Qu, A., Li, R. (2006). Quadratic inference functions for varying-coefficient models with longitudinal data. Biometrics, 62(2), 379–391.
Qu, A., Lindsay, B. G., Li, B. (2000). Improving generalised estimating equations using quadratic inference functions. Biometrika, 87(4), 823–836.
Şentürk, D., Müller, H.-G. (2008). Generalized varying coefficient models for longitudinal data. Biometrika, 95(3), 653–666.
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B, 58(1), 267–288.
Van der Vaart, A. W. (2000). Asymptotic statistics. Cambridge, UK: Cambridge University Press.
Verhasselt, A. (2014). Generalized varying coefficient models: a smooth variable selection technique. Statistica Sinica, 24(1), 147–171.
Wang, H. J., Zhu, Z., Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. The Annals of Statistics, 37(6), 3841–3866.
Wang, H., Xia, Y. (2009). Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association, 104(486), 747–757.
Wang, L., Li, H., Huang, J. Z. (2008). Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. Journal of the American Statistical Association, 103(484), 1556–1569.
Wang, N., Carroll, R. J., Lin, X. (2005). Efficient semiparametric marginal estimation for longitudinal/clustered data. Journal of the American Statistical Association, 100(469), 147–157.
Xue, L., Qu, A., Zhou, J. (2010). Consistent model selection for marginal generalized additive model for correlated data. Journal of the American Statistical Association, 105(492), 1518–1530.
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(1), 49–67.
Zhang, C.-H. (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38(2), 894–942.
Zhao, W., Zhang, R., Lv, Y., Liu, J. (2013). Variable selection of the quantile varying coefficient regression models. Journal of the Korean Statistical Society, 42(3), 343–358.
Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American statistical association, 101(476), 1418–1429.
Zou, H., Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Annals of statistics, 36(4), 1509–1533.
We sincerely thank the Editor/Professor Hironori Fujisawa, an associate editor and two anonymous reviewers, for their insightful comments that have led to significant improvement of the paper. Zhao’s research is supported in part by National Social Science Foundation of China (15BTJ027). Zhang’s research is supported by the National Natural Science Foundation of China Grant 11671374 and 71631006. Heng Lian’s research is partially supported by City University of Hong Kong Start-up Grant 7200521 and RGC General Research Fund 11301718.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Zhao, W., Zhang, W. & Lian, H. Marginal quantile regression for varying coefficient models with longitudinal data. Ann Inst Stat Math 72, 213–234 (2020) doi:10.1007/s10463-018-0684-7
- Longitudinal data
- Quadratic inference function
- Quantile regression
- Varying coefficient model