Empirical Likelihood Local Polynomial Regression Analysis of Clustered Data


Liugen Xue, College of Applied Sciences, Beijing University of Technology, Beijing 100124, P.R. China.
E-mail: lgxue@bjut.edu.cn


Abstract.  In this article, a naive empirical likelihood ratio is constructed for a non-parametric regression model with clustered data, by combining the empirical likelihood method and local polynomial fitting. The maximum empirical likelihood estimates for the regression functions and their derivatives are obtained. The asymptotic distributions for the proposed ratio and estimators are established. A bias-corrected empirical likelihood approach to inference for the parameters of interest is developed, and the residual-adjusted empirical log-likelihood ratio is shown to be asymptotically chi-squared. These results can be used to construct a class of approximate pointwise confidence intervals and simultaneous bands for the regression functions and their derivatives. Owing to our bias correction for the empirical likelihood ratio, the accuracy of the obtained confidence region is not only improved, but also a data-driven algorithm can be used for selecting an optimal bandwidth to estimate the regression functions and their derivatives. A simulation study is conducted to compare the empirical likelihood method with the normal approximation-based method in terms of coverage accuracies and average widths of the confidence intervals/bands. An application of this method is illustrated using a real data set.