Empirical likelihood for quantile regression models with longitudinal data. (English) Zbl 1204.62072
Summary: We develop two empirical likelihood-based inference procedures for longitudinal data under the framework of quantile regression. The proposed methods avoid estimating the unknown error density function and the intra-subject correlation involved in the asymptotic covariance matrix of the quantile estimators. By appropriately smoothing the quantile score function, the empirical likelihood approach is shown to have a higher-order accuracy through the Bartlett correction. The proposed methods exhibit finite-sample advantages over the normal approximation-based and bootstrap methods in a simulation study and the analysis of a longitudinal ophthalmology data set.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 65C60 | Computational problems in statistics (MSC2010) |
| 62G05 | Nonparametric estimation |
| 62G09 | Nonparametric statistical resampling methods |
| 62G15 | Nonparametric tolerance and confidence regions |
Keywords:
Bartlett correction; confidence region; estimating equation; hypothesis test; kernel smoothing; quantile regression; ophtalmologyReferences:
| [1] | Brown, B. M.; Wang, Y. G., Standard errors and covariance matrices for smoothed rank estimators, Biometrika, 92, 149-158 (2005) · Zbl 1068.62037 |
| [2] | Chen, L.; Wei, L. J.; Parzen, M., Quantile regression for correlated observations, (Lin, D.; Heagerty, P., Proceedings of the Second Seattle Symposium in Biostatistics: Analysis of Correlated Data (2003), Springer: Springer New York) · Zbl 1390.62109 |
| [3] | Chen, J.; Sitter, R. R.; Wu, C., Using empirical likelihood methods to obtain range restricted weights in regression estimators for surveys, Biometrika, 89, 230-237 (2002) · Zbl 0997.62008 |
| [4] | Chen, S. X.; Cui, H., On Bartlett correction of empirical likelihood in the presence of nuisance parameters, Biometrika, 93, 215-220 (2006) · Zbl 1152.62325 |
| [5] | Chen, S. X.; Hall, P., Smoothed empirical likelihood confidence intervals for quantiles, Ann. Statist., 22, 1166-1181 (1993) · Zbl 0786.62053 |
| [6] | Chen, S. X.; Wong, C. M., Smoothed block empirical likelihood for quantiles of weakly dependent process, Statist. Sinica, 19, 71-81 (2009) · Zbl 1153.62022 |
| [7] | DiCiccio, T. J.; Hall, P.; Romano, J. P., Empirical likelihood is Bartlett-correctable, Ann. Statist., 19, 1053-1061 (1991) · Zbl 0725.62042 |
| [8] | Geraci, M.; Bottai, M., Quantile regression for longitudinal data using the asymmetric Laplace distribution, Biostatistics, 8, 140-154 (2007) · Zbl 1170.62380 |
| [9] | He, X.; Shao, Q. M., A general Bahadur representation of M-estimators and its application to linear regression with nonstochatic designs, Ann. Statist., 24, 2608-2630 (1996) · Zbl 0867.62012 |
| [10] | Horowitz, J. L., Bootstrap methods for median regression models, Econometrica, 66, 1327-1351 (1998) · Zbl 1056.62517 |
| [11] | Jung, S., Quasi-likelihood for median regression models, J. Amer. Statist. Assoc., 91, 251-257 (1996) · Zbl 0871.62060 |
| [12] | Koenker, R., Quantile Regression (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1111.62037 |
| [13] | Koenker, R.; Bassett, B., Regression quantiles, Econometrica, 46, 33-50 (1978) · Zbl 0373.62038 |
| [14] | Lipsitz, S. R.; Fitzmaurice, G. M.; Molenberghs, G.; Zhao, L. P., 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 (1997) · Zbl 0908.62114 |
| [15] | McCullagh, P., Tensor Methods in Statistics (1987), Chapman & Hall: Chapman & Hall London · Zbl 0732.62003 |
| [16] | Otsu, T., Conditional empirical likelihood estimation and inference for quantile regression models, J. Econometrics, 142, 508-538 (2008) · Zbl 1418.62165 |
| [17] | Owen, A., Empirical likelihood ratio confidence intervals for a single functional, Biometrika, 75, 237-249 (1988) · Zbl 0641.62032 |
| [18] | Owen, A., Empirical likelihood ratio confidence regions, Ann. Statist., 18, 90-120 (1990) · Zbl 0712.62040 |
| [19] | Qin, G.; Tsao, M., Empirical likelihood inference for median regression models for censored survival data, J. Multivariate Anal., 85, 416-430 (2003) · Zbl 1016.62112 |
| [20] | Qin, J.; Lawless, J., Empirical likelihood and general estimating equations, Ann. Statist., 22, 300-325 (1994) · Zbl 0799.62049 |
| [21] | Song, P. K.; Tan, M., Marginal models for longitudinal continuous proportional data, Biometrics, 56, 496-502 (2000) · Zbl 1060.62667 |
| [22] | Subramanian, S., Censored median regression and profile empirical likelihood, Statist. Methodol., 4, 493-503 (2007) · Zbl 1248.62059 |
| [23] | Wang, Y. G.; Shao, Q. X.; Zhu, M., Quantile regression without the curse of unsmoothness, Comput. Statist. Data Anal., 53, 3696-3705 (2009) · Zbl 1453.62241 |
| [24] | Welsh, A. H.; Zhou, X. H., Estimating the retransformed mean in a heteroscedastic two-part model, J. Statist. Plann. Inference, 136, 860-881 (2006) · Zbl 1079.62039 |
| [25] | Whang, Y. J., Smoothed empirical likelihood methods for quantile regression models, Econometric Theory, 22, 173-205 (2006) · Zbl 1138.62017 |
| [26] | Xue, L. G.; Zhu, L. X., Empirical likelihood for a varying coefficient model with longitudinal data, J. Amer. Statist. Assoc., 102, 642-654 (2007) · Zbl 1172.62306 |
| [27] | Yin, G.; Cai, J., Quantile regression models with multivariate failure time data, Biometrics, 61, 151-161 (2005) · Zbl 1077.62086 |
| [28] | You, J.; Chen, G.; Zhou, Y., Block empirical likelihood for longitudinal partially linear regression models, Canad. J. Statist., 79, 79-96 (2006) · Zbl 1096.62033 |
| [29] | Zhao, Y.; Chen, F., Empirical likelihood inference for censored median regression model via nonparametric kernel estimation, J. Multivariate Anal., 99, 215-231 (2008) · Zbl 1139.62059 |
| [30] | Zhou, M., Bathke, A., Kim, M., 2009. Empirical likelihood analysis for the heteroscedastic accelerated failure time model, manuscript.; Zhou, M., Bathke, A., Kim, M., 2009. Empirical likelihood analysis for the heteroscedastic accelerated failure time model, manuscript. · Zbl 1534.62149 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
