×

zbMATH — the first resource for mathematics

Empirical likelihood inference in partially linear single-index models for longitudinal data. (English) Zbl 1181.62034
Summary: The empirical likelihood method is especially useful for constructing confidence intervals or regions of parameters of interest. Yet, the technique cannot be directly applied to partially linear single-index models for longitudinal data due to the within-subject correlation. In this paper, a bias-corrected block empirical likelihood (BCBEL) method is suggested to study the models by accounting for the within-subject correlation. BCBEL shares some desired features: unlike any normal approximation based method for confidence region, the estimation of parameters with the iterative algorithm is avoided and a consistent estimator of the asymptotic covariance matrix is not needed. Because of bias correction, the BCBEL ratio is asymptotically chi-squared, and hence it can be directly used to construct confidence regions of the parameters without any extra Monte Carlo approximation that is needed when bias correction is not applied. The proposed method can naturally be applied to deal with pure single-index models and partially linear models for longitudinal data. Some simulation studies are carried out and an example in epidemiology is given for illustration.

MSC:
62G05 Nonparametric estimation
62J05 Linear regression; mixed models
62G15 Nonparametric tolerance and confidence regions
62G20 Asymptotic properties of nonparametric inference
62G08 Nonparametric regression and quantile regression
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chiou, J.M.; Müller, H.G., Estimated estimating equations: semiparametric inference for clustered and longitudinal data, J. roy. statist. soc. B, 67, 4, 531-553, (2005) · Zbl 1095.62046
[2] Zeger, S.L.; Diggle, P.J., Semiparametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters, Biometrics, 50, 689-699, (1994) · Zbl 0821.62093
[3] Lin, X.H.; Carrol, R.J., Semiparametric regression for clustered data using generalized estimating equations, J. amer. statist. assoc., 96, 1045-1056, (2001) · Zbl 1072.62566
[4] He, X.M.; Zhu, Z.Y.; Fung, W.K., Estimation in a semiparametric model for longitudinal data with unspecified dependence structure, Biometrika, 89, 3, 579-590, (2002) · Zbl 1036.62035
[5] Fan, J.Q.; Li, R.Z., New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis, J. amer. statist. assoc., 99, 710-723, (2004) · Zbl 1117.62329
[6] W.M. Qian, Method of estimation in statistical models for longitudinal data. Ph.D. Thesis, Tongji University, Shanghai, 2003
[7] Sun, X.Q.; You, J.H., Iterative weighted partial spline least squares estimation in semiparametric modeling of longitudinal data, Sci. China, ser. A, 46, 5, 724-735, (2003) · Zbl 1130.62326
[8] You, J.H.; Chen, G.M.; Zhou, Y., Block empirical likelihood for longitudinal partially linear regression models, Canadian J. statist., 34, 1, 79-96, (2006) · Zbl 1096.62033
[9] Xue, L.G.; Zhu, L.X., Empirical likelihood semiparametric regression analysis for longitudinal data, Biometrika, 94, 921-937, (2007) · Zbl 1156.62324
[10] Fan, J.Q.; Huang, T.; Li, R.Z., Analysis of longitudinal data with semiparametric estimation of covariance function, J. amer. statist. assoc., 102, 632-641, (2007) · Zbl 1172.62323
[11] Li, G.R.; Tian, P.; Xue, L.G., Generalized empirical likelihood inference in semiparametric regression model for longitudinal data, Acta math. sin. (engl. ser.), 24, 12, 2029-2040, (2008) · Zbl 1151.62316
[12] Carroll, R.J.; Fan, J.Q.; Gijbels, I.; Wand, M.P., Generalized partially linear single-index models, J. amer. statist. assoc., 92, 477-489, (1997) · Zbl 0890.62053
[13] Xue, L.G.; Zhu, L.X., Empirical likelihood confidence regions of the parameters in a partially linear single-index model, Sci. China, ser. A, 48, 10, 1333-1348, (2005) · Zbl 1112.62027
[14] Zhu, L.X.; Xue, L.G., Empirical likelihood confidence regions in a partially linear single-index model, J. roy. statist. soc. B, 68, 549-570, (2006) · Zbl 1110.62055
[15] Yu, Y.; Ruppert, D., Penalized spline estimation for partially linear single-index models, J. amer. statist. assoc., 97, 1042-1054, (2002) · Zbl 1045.62035
[16] Xia, Y.; Härdle, W., Semi-parametric estimation of partially linear single-index models, J. multivariate anal., 97, 1162-1184, (2006) · Zbl 1089.62050
[17] Owen, A.B., Empirical likelihood ratio confidence intervals for a single function, Biometrika, 75, 2, 237-249, (1988) · Zbl 0641.62032
[18] Owen, A.B., Empirical likelihood ratio confidence regions, Ann. statist., 18, 90-120, (1990) · Zbl 0712.62040
[19] Owen, A.B., Empirical likelihood, (2001), Chapman & Hall/CRC New York · Zbl 0989.62019
[20] Kitamura, Y., Empirical likelihood methods with weakly dependent processes, Ann. statist., 25, 2084-2102, (1997) · Zbl 0881.62095
[21] 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
[22] Liang, K.Y.; Zeger, S.L., Longitudinal data analysis using generalized linear model, Biometrika, 73, 13-22, (1986) · Zbl 0595.62110
[23] Fan, J.Q.; Gijbels, I., Local polynomial modeling and its applications, (1996), Chapman and Hall London · Zbl 0873.62037
[24] Lin, X.H.; Carrol, R.J., Nonparametric function estimation for clustered data when the predictor is measured without/with error, J. amer. statist. assoc., 95, 520-534, (2000) · Zbl 0995.62043
[25] Xue, L.G.; Zhu, L.X., Empirical likelihood for single-index models, J. multivariate anal., 97, 6, 1295-1312, (2006) · Zbl 1099.62045
[26] Li, K.C., Slice inverse regression for dimension reduction, J. amer. statist. assoc., 86, 316-327, (1991) · Zbl 0742.62044
[27] Kaslow, R.A.; Ostrow, D.G.; Detels, R.; Phair, J.P.; Polk, B.F.; Rinaldo, C.R., The multicenter AIDS cohort study: rationale, organization and selected characteristics of the participants, Am. J. epidemiol., 126, 310-318, (1987)
[28] Wu, C.O.; Chiang, C.T.; Hoover, D.R., Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data, J. amer. statist. assoc., 93, 1388-1402, (1998) · Zbl 1064.62523
[29] Huang, J.Z.; Wu, C.O.; Zhou, L., Varying-coefficient models and basis function approximations for the analysis of repeated measurements, Biometrika, 89, 111-128, (2002) · Zbl 0998.62024
[30] Fan, J.Q.; Zhang, J., Two-step estimation of functional linear models with applications to longitudinal data, J. roy. statist. soc. B, 62, 303-322, (2000)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.