Block empirical likelihood inference for semiparametric varying-coeffcient partially linear errors-in-variables models with longitudinal data is investigated. We apply the block empirical likelihood procedure to accommodate the within-group correlation of the longitudinal data. The block empirical log-likelihood ratio statistic for the parametric component is suggested. And the nonparametric version of Wilk’s theorem is derived under mild conditions. Simulations are carried out to access the performance of the proposed procedure. 1. Introduction For longitudinal data, we consider semiparametric varying-coefficient partially linear model which has the following form: where is the response variable, , , and are regressors, is a -dimensional vector of unknown parameters, is a -dimensional vector of smooth functions of time , and is a zero-mean stochastic process. Due to the curse of dimensionality, for simplicity, we assume that is univariate. Obviously, model (1) contains many usual parametric, nonparametric, and semiparametric models. Model (1) has been studied by many authors. Zhang et al. [1] suggested a two-step method for estimating it. Li et al. [2] suggested a local least-squares procedure with a kernel weight function. Fan and Huang [3] developed a profile least-squares technique for estimating parametric model. You and Zhou [4] and Huang and Zhang [5] suggested the estimator of the parametric and nonparametric models, respectively. Fan et al. [6] proposed a semiparametric estimation of the working correlation matrix and applied a profile weighted least-squares approach. However, in many practical situations, these variables are often measured with error. In this paper, we consider this case where the variable is measured with additive error and both and are measured exactly. That is, cannot be observed, but an unbiased measure of , denoted by , can be obtained as follows: where is the measurement error, which is independent of ( , , , ), with mean zero and covariance matrix . We can assume that is known. If is unknown, we estimate it by repeatedly measuring by Liang et al. [7]. For errors-in-variables models (1) and (2), Liang et al. [8] developed a profile least-squares procedure to estimate the parametric component and derived the asymptotic normality of the resulting estimator. The empirical likelihood, which is a nonparametric approach for constructing confidence regions, was introduced by Owen [9] and has many nice statistical properties (see Owen [10]). Owen [11] applied empirical likelihood to linear regression models and Kolaczyk [12]
References
[1]
W. Zhang, S.-Y. Lee, and X. Song, “Local polynomial fitting in semivarying coefficient model,” Journal of Multivariate Analysis, vol. 82, no. 1, pp. 166–188, 2002.
[2]
Q. Li, C. J. Huang, D. Li, and T.-T. Fu, “Semiparametric smooth coefficient models,” Journal of Business & Economic Statistics, vol. 20, no. 3, pp. 412–422, 2002.
[3]
J. Fan and T. Huang, “Profile likelihood inferences on semiparametric varying-coefficient partially linear models,” Bernoulli, vol. 11, no. 6, pp. 1031–1057, 2005.
[4]
J. You and Y. Zhou, “Empirical likelihood for semiparametric varying-coefficient partially linear regression models,” Statistics & Probability Letters, vol. 76, no. 4, pp. 412–422, 2006.
[5]
Z. Huang and R. Zhang, “Empirical likelihood for nonparametric parts in semiparametric varying-coefficient partially linear models,” Statistics & Probability Letters, vol. 79, no. 16, pp. 1798–1808, 2009.
[6]
J. Fan, T. Huang, and R. Li, “Analysis of longitudinal data with semiparametric estimation of convariance function,” Journal of the American Statistical Association, vol. 102, no. 478, pp. 632–641, 2007.
[7]
H. Liang, W. H？rdle, and R. J. Carroll, “Estimation in a semiparametric partially linear errors-in-variables model,” The Annals of Statistics, vol. 27, no. 5, pp. 1519–1535, 1999.
[8]
H. Liang, S. W. Thurston, D. Ruppert, T. Apanasovich, and R. Hauser, “Additive partial linear models with measurement errors,” Biometrika, vol. 95, no. 3, pp. 667–678, 2008.
[9]
A. B. Owen, “Empirical likelihood ratio confidence intervals for a single functional,” Biometrika, vol. 75, no. 2, pp. 237–249, 1988.
[10]
A. Owen, “Empirical likelihood ratio confidence regions,” The Annals of Statistics, vol. 18, no. 1, pp. 90–120, 1990.
[11]
A. Owen, “Empirical likelihood for linear models,” The Annals of Statistics, vol. 19, no. 4, pp. 1725–1747, 1991.
[12]
E. D. Kolaczyk, “Empirical likelihood for generalized linear models,” Statistica Sinica, vol. 4, no. 1, pp. 199–218, 1994.
[13]
L. Xue and L. Zhu, “Empirical likelihood for a varying coefficient model with longitudinal data,” Journal of the American Statistical Association, vol. 102, no. 478, pp. 642–654, 2007.
[14]
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, pp. 907–923, 2009.
[15]
H. Yang and T. Li, “Empirical likelihood for semiparametric varying coefficient partially linear models with longitudinal data,” Statistics & Probability Letters, vol. 80, no. 2, pp. 111–121, 2010.
[16]
X. Hu, Z. Wang, and Z. Zhao, “Empirical likelihood for semiparametric varying-coefficient partially linear errors-in-variables models,” Statistics & Probability Letters, vol. 79, no. 8, pp. 1044–1052, 2009.
[17]
X. Wang, G. Li, and L. Lin, “Empirical likelihood inference for semi-parametric varying-coefficient partially linear EV models,” Metrika, vol. 73, no. 2, pp. 171–185, 2011.
[18]
G.-L. Fan, H.-X. Xu, and H.-Y. Liang, “Empirical likelihood inference for partially time-varying coefficient errors-in-variables models,” Electronic Journal of Statistics, vol. 6, pp. 1040–1058, 2012.
[19]
G.-L. Fan, H.-Y. Liang, and J.-F. Wang, “Statistical inference for partially time-varying coefficient errors-in-variables models,” Journal of Statistical Planning and Inference, vol. 143, no. 3, pp. 505–519, 2013.
[20]
J. You, G. Chen, and Y. Zhou, “Block empirical likelihood for longitudinal partially linear regression models,” The Canadian Journal of Statistics, vol. 34, no. 1, pp. 79–96, 2006.
[21]
X. Hu, “The block empirical likelihood method of the semivarying coefficient model with application to longitudinal data,” Communications in Statistics. Theory and Methods, vol. 40, no. 8, pp. 1342–1351, 2011.
[22]
W. A. Fuller, Measurement Error Models, John Wiley & Sons, New York, NY, USA, 1987.
[23]
Y. P. Mack and B. W. Silverman, “Weak and strong uniform consistency of kernel regression estimates,” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 61, no. 3, pp. 405–415, 1982.
[24]
J. Shi and T.-S. Lau, “Empirical likelihood for partially linear models,” Journal of Multivariate Analysis, vol. 72, no. 1, pp. 132–148, 2000.
