Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically χq+r2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi ^2_{q+r}$$\end{document}, where q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} and r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document} are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.