Inference Based on Empirical Likelihood for Varying Coefficient Model with Random Effect
ABSTRACT
In this article, we develop a statistical inference technique for the unknown coefficient functions in the varying coeffi- cient model with random effect. A residual-adjusted block empirical likelihood (RABEL) method is suggested to inves- tigate the model by taking the within-subject correlation into account. Due to the residual adjustment, the proposed RABEL is asymptotically chi-squared distribution. We illustrate the large sample performance of the proposed method via
Cite this paper
W. Li and L. Xue, "Inference Based on Empirical Likelihood for Varying Coefficient Model with Random Effect," Open Journal of Statistics, Vol. 3 No. 6, 2013, pp. 52-59. doi: 10.4236/ojs.2013.36A006.
W. Li and L. Xue, "Inference Based on Empirical Likelihood for Varying Coefficient Model with Random Effect," Open Journal of Statistics, Vol. 3 No. 6, 2013, pp. 52-59. doi: 10.4236/ojs.2013.36A006.
References
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http://dx.doi.org/10.1016/j.csda.2011.11.007 [10] A. B. Owen, “Empirical Likelihood Ratio Confidence Intervals for a Single Functional,” Biometrika, Vol. 75, No. 2, 1988, pp. 237-249.
http://dx.doi.org/10.1093/biomet/75.2.237 [11] A. Owen, “Empirical Likelihood Ratio Confidence Regions,” Annals of Statistics, Vol. 18, No. 1, 1990, pp. 90120. http://dx.doi.org/10.1214/aos/1176347494 [12] A. Owen, “Empirical Likelihood for Linear Models,” Annals of Statistics, Vol. 19, No. 4, 1991, pp. 1725-1747.
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http://dx.doi.org/10.1016/j.jmva.2010.03.005 [18] J. Fan and R. Li, “New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis,” Journal of the American Statistical Association, Vol. 99, No. 467, 2004, pp. 710-723.
http://dx.doi.org/10.1198/016214504000001060 [19] H. G. M¨uller and J. M. Chiou, “Nonparametric QuasiLikelihood,” Annals of Statistics, Vol. 27, No. 1, 1999, pp. 36-64. http://dx.doi.org/10.1214/aos/1018031100 [20] D. A. Harville, “Matrix Algebra from a Statistician’s Perspective,” Springer, New York, 1997.
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http://dx.doi.org/10.1002/9780470226797 [22] Y. Li, “Efficient Semiparametric Regression for Longitudinal Data with Nonparametric Covariance Estimation,” Biometrika, Vol. 98, No. 2, 2011, pp. 355-370.
http://dx.doi.org/10.1093/biomet/asq080
[1] C. T. Chiang, J. A. Rice and C. O. Wu, “Smoothing Spline Estimation for Varying Coefficient Models with Repeatedly Measured Dependent Variables,” Journal of the American Statistical Association, Vol. 96, No. 454, 2001, pp. 605-619. [2] A. Qu and R. Li, “Quadratic Inference Functions for Varying Coefficient Models with Longitudinal Data,” Biometrics, Vol. 62, No. 2, 2006, pp. 379-391.
http://dx.doi.org/10.1111/j.1541-0420.2005.00490.x [3] L. G. Xue and L. X. Zhu, “Empirical Likelihood for a Varying Coefficient Model with Longitudinal Data,” Journal of the American Statistical Association, Vol. 102, No. 478, 2007, pp. 642-652.
http://dx.doi.org/10.1198/016214507000000293 [4] H. J. Wang, Z. Zhu and J. Zhou, “Quantile Regression in Partially Linear Varying Coefficient Models,” Annals of Statistics, Vol. 37, No. 6B, 2009, pp. 3841-3866. [5] R. A. Kaslow, D. G. Ostrow, R. Detels, J. P. Phair, B. F. Polk and C. R. Rinaldo, “The Multicenter AIDS Cohort Study: Rationale, Organization and Selected Characteristics of the Participants,” American Journal of Epidemiology, Vol. 126, No. 2, 1987, pp. 310-318.
http://dx.doi.org/10.1093/aje/126.2.310 [6] Q. Li and A. Ullah, “Estimating Partially Linear Panel Data Models with One-way Error Components,” Econometric Reviews, Vol. 17, No. 2, 1998, pp. 145-166.
http://dx.doi.org/10.1080/07474939808800409 [7] C. Gu and P. Ma, “Optimal Smoothing in Nonparametric Mixed Effect Models,” Annals of Statistics, Vol. 33, No. 3, 2005, pp. 1357-1379.
http://dx.doi.org/10.1214/009053605000000110 [8] J. You, X. Zhou and Y. Zhou, “Statistical Inference for Panel Data Semiparametric Partially Linear Regression Models with Heteroscedastic Errors,” Journal of Multivariate Analysis, Vol. 101, No. 5, 2010, pp. 1079-1101.
http://dx.doi.org/10.1016/j.jmva.2010.01.003 [9] Z. Pang and L. G. Xue, “Estimation for the Single-index Models with Random Effects,” Computational Statistics & Data Analysis, Vol. 56, No. 6, 2012, pp. 1837-1853.
http://dx.doi.org/10.1016/j.csda.2011.11.007 [10] A. B. Owen, “Empirical Likelihood Ratio Confidence Intervals for a Single Functional,” Biometrika, Vol. 75, No. 2, 1988, pp. 237-249.
http://dx.doi.org/10.1093/biomet/75.2.237 [11] A. Owen, “Empirical Likelihood Ratio Confidence Regions,” Annals of Statistics, Vol. 18, No. 1, 1990, pp. 90120. http://dx.doi.org/10.1214/aos/1176347494 [12] A. Owen, “Empirical Likelihood for Linear Models,” Annals of Statistics, Vol. 19, No. 4, 1991, pp. 1725-1747.
http://dx.doi.org/10.1214/aos/1176348368 [13] P. Zhao and L. Xue, “Empirical Likelihood Inferences for Semiparametric Varying Coefficient Partially Linear Errors in Variables Models with Longitudinal Data,” Journal of Nonparametric Statistics, Vol. 21, No. 7, 2009, pp. 907-923. http://dx.doi.org/10.1080/10485250902980576 [14] G. R. Li, P. Tian and L. G. Xue, “Generalized Empirical Likelihood Inference in Semiparametric Regression Model for Longitudinal Data,” Acta Mathematica Sinica, Vol. 24, No. 12, 2008, pp. 2029-2040.
http://dx.doi.org/10.1007/s10114-008-6434-7 [15] K. Y. Liang and S. L. Zeger, “Longitudinal Data Analysis Using Generalised Linear Models,” Biometrika, Vol. 73, No. 1, 1986, pp. 12-22.
http://dx.doi.org/10.1093/biomet/73.1.13 [16] J. You, G. Chen and Y. Zhou, “Block Empirical Likelihood for Longitudinal Partially Linear Regression Models,” Canadian Journal of Statistics, Vol. 34, No. 1, 2006, pp. 79-96.
http://dx.doi.org/10.1002/cjs.5550340107 [17] P. Zhao and L. Xue, “Variable Selection for Semiparametric Varying Coefficient Partially Linear Errors in Variables Models,” Journal of Multivariate Analysis, Vol. 101, No. 8, 2010, pp. 1872-1883.
http://dx.doi.org/10.1016/j.jmva.2010.03.005 [18] J. Fan and R. Li, “New Estimation and Model Selection Procedures for Semiparametric Modeling in Longitudinal Data Analysis,” Journal of the American Statistical Association, Vol. 99, No. 467, 2004, pp. 710-723.
http://dx.doi.org/10.1198/016214504000001060 [19] H. G. M¨uller and J. M. Chiou, “Nonparametric QuasiLikelihood,” Annals of Statistics, Vol. 27, No. 1, 1999, pp. 36-64. http://dx.doi.org/10.1214/aos/1018031100 [20] D. A. Harville, “Matrix Algebra from a Statistician’s Perspective,” Springer, New York, 1997.
http://dx.doi.org/10.1007/b98818 [21] G. A. F. Seber, “A Matrix Handbook for Statisticians,” John Wiley & Sons, Hoboken, 2007.
http://dx.doi.org/10.1002/9780470226797 [22] Y. Li, “Efficient Semiparametric Regression for Longitudinal Data with Nonparametric Covariance Estimation,” Biometrika, Vol. 98, No. 2, 2011, pp. 355-370.
http://dx.doi.org/10.1093/biomet/asq080