The Annals of Statistics
- Ann. Statist.
- Volume 43, Number 4 (2015), 1682-1715.
Semiparametric GEE analysis in partially linear single-index models for longitudinal data
Jia Chen, Degui Li, Hua Liang, and Suojin Wang
Full-text: Open access
Abstract
In this article, we study a partially linear single-index model for longitudinal data under a general framework which includes both the sparse and dense longitudinal data cases. A semiparametric estimation method based on a combination of the local linear smoothing and generalized estimation equations (GEE) is introduced to estimate the two parameter vectors as well as the unknown link function. Under some mild conditions, we derive the asymptotic properties of the proposed parametric and nonparametric estimators in different scenarios, from which we find that the convergence rates and asymptotic variances of the proposed estimators for sparse longitudinal data would be substantially different from those for dense longitudinal data. We also discuss the estimation of the covariance (or weight) matrices involved in the semiparametric GEE method. Furthermore, we provide some numerical studies including Monte Carlo simulation and an empirical application to illustrate our methodology and theory.
Article information
Source
Ann. Statist., Volume 43, Number 4 (2015), 1682-1715.
Dates
Received: May 2014
Revised: February 2015
First available in Project Euclid: 17 June 2015
Permanent link to this document
https://projecteuclid.org/euclid.aos/1434546219
Digital Object Identifier
doi:10.1214/15-AOS1320
Mathematical Reviews number (MathSciNet)
MR3357875
Zentralblatt MATH identifier
1317.62036
Subjects
Primary: 62G09: Resampling methods 62H99: None of the above, but in this section 62G99: None of the above, but in this section
Keywords
Efficiency GEE local linear smoothing longitudinal data semiparametric estimation single-index models
Citation
Chen, Jia; Li, Degui; Liang, Hua; Wang, Suojin. Semiparametric GEE analysis in partially linear single-index models for longitudinal data. Ann. Statist. 43 (2015), no. 4, 1682--1715. doi:10.1214/15-AOS1320. https://projecteuclid.org/euclid.aos/1434546219
References
- Braun-Fahrländer, C., Ackermann-Liebrich, U., Schwartz, J., Gnehm, H. P., Rutishauser, M. and Wanner, H. U. (1992). Air pollution and respiratory symptoms in preschool children. Am. Rev. Respir. Dis. 145 42–47.
- Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489.Mathematical Reviews (MathSciNet): MR1467842
Zentralblatt MATH: 0890.62053
Digital Object Identifier: doi:10.1080/01621459.1997.10474001 - Chen, L.-H., Cheng, M.-Y. and Peng, L. (2009). Conditional variance estimation in heteroscedastic regression models. J. Statist. Plann. Inference 139 236–245.Mathematical Reviews (MathSciNet): MR2474001
Zentralblatt MATH: 1149.62032
Digital Object Identifier: doi:10.1016/j.jspi.2008.04.020 - Chen, J., Gao, J. and Li, D. (2013a). Estimation in single-index panel data models with heterogeneous link functions. Econometric Rev. 32 928–955.Mathematical Reviews (MathSciNet): MR3041108
Digital Object Identifier: doi:10.1080/07474938.2012.690687 - Chen, J., Gao, J. and Li, D. (2013b). Estimation in partially linear single-index panel data models with fixed effects. J. Bus. Econom. Statist. 31 315–330.Mathematical Reviews (MathSciNet): MR3173684
- Chen, J., Li, D., Liang, H. and Wang, S. (2015). Supplement to “Semiparametric GEE analysis in partially linear single-index models for longitudinal data.” DOI:10.1214/15-AOS1320SUPP.
- Diggle, P. J., Heagerty, P. J., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data, 2nd ed. Oxford Univ. Press, Oxford.Mathematical Reviews (MathSciNet): MR2049007
- Dockery, D. W., Speizer, F. E., Stram, D. O., Ware, J. H., Spengler, J. D. and Ferris, B. G. Jr. (1989). Effects of inhalable particles on respiratory health of children. Am. Rev. Respir. Dis. 139 587–594.
- Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall, London.
- Fan, J. and Huang, T. (2005). Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli 11 1031–1057.Mathematical Reviews (MathSciNet): MR2189080
Digital Object Identifier: doi:10.3150/bj/1137421639
Project Euclid: euclid.bj/1137421639 - Fan, J., Huang, T. and Li, R. (2007). Analysis of longitudinal data with semiparametric estimation of convariance function. J. Amer. Statist. Assoc. 102 632–641.Mathematical Reviews (MathSciNet): MR2370857
Digital Object Identifier: doi:10.1198/016214507000000095 - Fan, J. and Li, R. (2004). New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. J. Amer. Statist. Assoc. 99 710–723.Mathematical Reviews (MathSciNet): MR2090905
Zentralblatt MATH: 1117.62329
Digital Object Identifier: doi:10.1198/016214504000001060 - Fan, J. and Wu, Y. (2008). Semiparametric estimation of covariance matrixes for longitudinal data. J. Amer. Statist. Assoc. 103 1520–1533.Mathematical Reviews (MathSciNet): MR2504201
Zentralblatt MATH: 1286.62030
Digital Object Identifier: doi:10.1198/016214508000000742 - Fan, J. and Yao, Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85 645–660.Mathematical Reviews (MathSciNet): MR1665822
Zentralblatt MATH: 0918.62065
Digital Object Identifier: doi:10.1093/biomet/85.3.645 - Gao, J. (2007). Nonlinear Time Series: Semiparametric and Nonparametric Methods. Chapman & Hall/CRC, Boca Raton, FL.Mathematical Reviews (MathSciNet): MR2297190
- Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.Mathematical Reviews (MathSciNet): MR2278365
Zentralblatt MATH: 1113.62073
Digital Object Identifier: doi:10.1214/009053606000000272
Project Euclid: euclid.aos/1152540756 - 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.
- 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.Mathematical Reviews (MathSciNet): MR1888349
Zentralblatt MATH: 0998.62024
Digital Object Identifier: doi:10.1093/biomet/89.1.111 - Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.Mathematical Reviews (MathSciNet): MR1230981
Zentralblatt MATH: 0816.62079
Digital Object Identifier: doi:10.1016/0304-4076(93)90114-K - Jiang, C.-R. and Wang, J.-L. (2011). Functional single index models for longitudinal data. Ann. Statist. 39 362–388.Mathematical Reviews (MathSciNet): MR2797850
Zentralblatt MATH: 1209.62073
Digital Object Identifier: doi:10.1214/10-AOS845
Project Euclid: euclid.aos/1291388379 - Kim, S. and Zhao, Z. (2013). Unified inference for sparse and dense longitudinal models. Biometrika 100 203–212.Mathematical Reviews (MathSciNet): MR3034333
Zentralblatt MATH: 1284.62536
Digital Object Identifier: doi:10.1093/biomet/ass050 - Kinney, P. L., Ware, J. H., Spengler, J. D., Dockery, D. W., Speizer, F. E. and Ferris, B. G. Jr. (1989). Short-term pulmonary function change in association with ozone levels. Am. Rev. Respir. Dis. 139 56–61.
- Leng, C. and Tang, C. Y. (2011). Improving variance function estimation in semiparametric longitudinal data analysis. Canad. J. Statist. 39 656–670.
- Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321–3351.Mathematical Reviews (MathSciNet): MR2766854
Zentralblatt MATH: 1204.62067
Digital Object Identifier: doi:10.1214/10-AOS813
Project Euclid: euclid.aos/1284988408 - Liang, K. Y. and Zeger, S. L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73 13–22.Mathematical Reviews (MathSciNet): MR836430
Zentralblatt MATH: 0595.62110
Digital Object Identifier: doi:10.1093/biomet/73.1.13 - Liang, H., Liu, X., Li, R. and Tsai, C.-L. (2010). Estimation and testing for partially linear single-index models. Ann. Statist. 38 3811–3836.Mathematical Reviews (MathSciNet): MR2766869
Zentralblatt MATH: 1204.62068
Digital Object Identifier: doi:10.1214/10-AOS835
Project Euclid: euclid.aos/1291126974 - Lin, X. and Carroll, R. J. (2006). Semiparametric estimation in general repeated measures problems. J. R. Stat. Soc. Ser. B. Stat. Methodol. 68 69–88.Mathematical Reviews (MathSciNet): MR2212575
Zentralblatt MATH: 1141.62026
Digital Object Identifier: doi:10.1111/j.1467-9868.2005.00533.x - Lin, D. Y. and Ying, Z. (2001). Semiparametric and nonparametric regression analysis of longitudinal data. J. Amer. Statist. Assoc. 96 103–126.Mathematical Reviews (MathSciNet): MR1952726
Zentralblatt MATH: 1015.62038
Digital Object Identifier: doi:10.1198/016214501750333018 - Lipfert, F. W. and Hammerstrom, T. (1992). Temporal patterns in air pollution and hospital admissions. Environ. Res. 59 374–399.
- Luo, W., Li, B. and Yin, X. (2014). On efficient dimension reduction with respect to a statistical functional of interest. Ann. Statist. 42 382–412.Mathematical Reviews (MathSciNet): MR3189490
Zentralblatt MATH: 1285.62072
Digital Object Identifier: doi:10.1214/13-AOS1195
Project Euclid: euclid.aos/1395234982 - Ma, S., Liang, H. and Tsai, C.-L. (2014). Partially linear single index models for repeated measurements. J. Multivariate Anal. 130 354–375.Mathematical Reviews (MathSciNet): MR3229543
Zentralblatt MATH: 1292.62060
Digital Object Identifier: doi:10.1016/j.jmva.2014.06.011 - Ma, Y. and Zhu, L. (2013). Doubly robust and efficient estimators for heteroscedastic partially linear single-index models allowing high dimensional covariates. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 305–322.Mathematical Reviews (MathSciNet): MR3021389
Digital Object Identifier: doi:10.1111/j.1467-9868.2012.01040.x - Peng, L. and Yao, Q. (2003). Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90 967–975.
- Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables, Oxford Science Publications. Oxford Univ. Press, New York.Mathematical Reviews (MathSciNet): MR1353441
- Pope, C. A. III (1991). Respiratory hospital admissions associated with $\mathrm{PM}_{1}0$ pollution in utah, salt lake, and cache valleys. Archives of Environmental Health: An International Journal 46 90–97.
- Pope, C. A. III, Bates, D. V. and Raizenne, M. E. (1995). Health effects of particulate air pollution: Time for reassessment? Environ. Health Perspect. 103 472–480.
- Wang, L. (2011). GEE analysis of clustered binary data with diverging number of covariates. Ann. Statist. 39 389–417.Mathematical Reviews (MathSciNet): MR2797851
Zentralblatt MATH: 1209.62138
Digital Object Identifier: doi:10.1214/10-AOS846
Project Euclid: euclid.aos/1291388380 - Wang, N., Carroll, R. J. and Lin, X. (2005). Efficient semiparametric marginal estimation for longitudinal/clustered data. J. Amer. Statist. Assoc. 100 147–157.Mathematical Reviews (MathSciNet): MR2156825
Zentralblatt MATH: 1117.62440
Digital Object Identifier: doi:10.1198/016214504000000629 - Wang, S., Qian, L. and Carroll, R. J. (2010). Generalized empirical likelihood methods for analyzing longitudinal data. Biometrika 97 79–93.Mathematical Reviews (MathSciNet): MR2594418
Zentralblatt MATH: 1183.62060
Digital Object Identifier: doi:10.1093/biomet/asp073 - Wang, J.-L., Xue, L., Zhu, L. and Chong, Y. S. (2010). Estimation for a partial-linear single-index model. Ann. Statist. 38 246–274.
- Wu, W. B. and Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika 90 831–844.
- Wu, H. and Zhang, J.-T. (2006). Nonparametric Regression Methods for Longitudinal Data Analysis. Wiley, Hoboken, NJ.
- Xia, Y. and Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. J. Multivariate Anal. 97 1162–1184.Mathematical Reviews (MathSciNet): MR2276153
Zentralblatt MATH: 1089.62050
Digital Object Identifier: doi:10.1016/j.jmva.2005.11.005 - Xia, Y., Tong, H. and Li, W. K. (1999). On extended partially linear single-index models. Biometrika 86 831–842.Mathematical Reviews (MathSciNet): MR1741980
Zentralblatt MATH: 0942.62109
Digital Object Identifier: doi:10.1093/biomet/86.4.831 - Xie, M. and Yang, Y. (2003). Asymptotics for generalized estimating equations with large cluster sizes. Ann. Statist. 31 310–347.Mathematical Reviews (MathSciNet): MR1962509
Zentralblatt MATH: 1018.62019
Digital Object Identifier: doi:10.1214/aos/1046294467
Project Euclid: euclid.aos/1046294467 - Yao, W. and Li, R. (2013). New local estimation procedure for a non-parametric regression function for longitudinal data. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 123–138.Mathematical Reviews (MathSciNet): MR3008274
Digital Object Identifier: doi:10.1111/j.1467-9868.2012.01038.x - Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577–590.Mathematical Reviews (MathSciNet): MR2160561
Zentralblatt MATH: 1117.62451
Digital Object Identifier: doi:10.1198/016214504000001745 - Yu, K. and Jones, M. C. (2004). Likelihood-based local linear estimation of the conditional variance function. J. Amer. Statist. Assoc. 99 139–144.Mathematical Reviews (MathSciNet): MR2054293
Zentralblatt MATH: 1089.62507
Digital Object Identifier: doi:10.1198/016214504000000133 - Yu, Y. and Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models. J. Amer. Statist. Assoc. 97 1042–1054.Mathematical Reviews (MathSciNet): MR1951258
Zentralblatt MATH: 1045.62035
Digital Object Identifier: doi:10.1198/016214502388618861 - Zhang, J.-T. and Chen, J. (2007). Statistical inferences for functional data. Ann. Statist. 35 1052–1079.Mathematical Reviews (MathSciNet): MR2341698
Zentralblatt MATH: 1129.62029
Digital Object Identifier: doi:10.1214/009053606000001505
Project Euclid: euclid.aos/1185303998 - Zhang, W., Fan, J. and Sun, Y. (2009). A semiparametric model for cluster data. Ann. Statist. 37 2377–2408.Mathematical Reviews (MathSciNet): MR2543696
Zentralblatt MATH: 1173.62030
Digital Object Identifier: doi:10.1214/08-AOS662
Project Euclid: euclid.aos/1247663759 - Zhang, W., Leng, C. and Tang, C. Y. (2015). A joint modelling approach for longitudinal studies. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 219–238.
Supplemental materials
- Supplement to “Semiparametric GEE analysis in partially linear single-index models for longitudinal data”. The supplement gives the proof of Theorem 3 and some technical lemmas that were used to prove the main results in Appendix B. It also includes some additional results of our simulation studies described in Section 5.Digital Object Identifier: doi:10.1214/15-AOS1320SUPP
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates
Zhang, Jun, Wang, Tao, Zhu, Lixing, and Liang, Hua, Electronic Journal of Statistics, 2012 - Efficient semiparametric estimation in generalized partially linear additive models for longitudinal/clustered data
Cheng, Guang, Zhou, Lan, and Huang, Jianhua Z., Bernoulli, 2014 - Efficient estimation in semivarying coefficient models for longitudinal/clustered data
Cheng, Ming-Yen, Honda, Toshio, and Li, Jialiang, The Annals of Statistics, 2016
- A dimension reduction based approach for estimation and variable selection in partially linear single-index models with high-dimensional covariates
Zhang, Jun, Wang, Tao, Zhu, Lixing, and Liang, Hua, Electronic Journal of Statistics, 2012 - Efficient semiparametric estimation in generalized partially linear additive models for longitudinal/clustered data
Cheng, Guang, Zhou, Lan, and Huang, Jianhua Z., Bernoulli, 2014 - Efficient estimation in semivarying coefficient models for longitudinal/clustered data
Cheng, Ming-Yen, Honda, Toshio, and Li, Jialiang, The Annals of Statistics, 2016 - Quantile regression in partially linear varying coefficient models
Wang, Huixia Judy, Zhu, Zhongyi, and Zhou, Jianhui, The Annals of Statistics, 2009 - Functional linear regression analysis for longitudinal data
Yao, Fang, Müller, Hans-Georg, and Wang, Jane-Ling, The Annals of Statistics, 2005 - Iterative estimating equations: Linear convergence and asymptotic properties
Jiang, Jiming, Luan, Yihui, and Wang, You-Gan, The Annals of Statistics, 2007 - Efficient estimation for generalized partially linear single-index models
Wang, Li and Cao, Guanqun, Bernoulli, 2018 - Fused kernel-spline smoothing for repeatedly measured outcomes in a generalized partially linear model with functional single index
Jiang, Fei, Ma, Yanyuan, and Wang, Yuanjia, The Annals of Statistics, 2015 - Semiparametric quantile estimation for varying coefficient partially linear measurement errors models
Zhang, Jun, Zhou, Yan, Cui, Xia, and Xu, Wangli, Brazilian Journal of Probability and Statistics, 2018 - Inference for single-index quantile regression models with profile optimization
Ma, Shujie and He, Xuming, The Annals of Statistics, 2016