Generalized Empirical Likelihood for Partially Linear Errors‐in‐Variables Regression Model With Longitudinal Data

In this paper, we study the generalized empirical likelihood inference for a partially linear model with measurement errors in both the covariates of parametric and nonparametric parts, on the basis of longitudinal data. The proposed generalized empirical likelihood approach considers within‐group correlations to estimate regression coefficients and does not involve direct estimation of nuisance parameters in the correlation matrix. At the true parameter, the empirical log$$ \log $$‐likelihood ratio is proved to be asymptotically chi‐square distributed under regularity conditions, and the corresponding confidence intervals are constructed. Next, the profile EL ratios for the parameter β$$ \beta $$ are given, and it is verified that these ratios follow the chi‐square distribution. Furthermore, the strong consistency and the convergence rate for the estimator of a nonparametric function are obtained, and the n$$ \sqrt{n} $$‐consistency of the estimator of σε2$$ {\sigma}_{\varepsilon}^2 $$ is verified. The performance of the proposed method is verified by numerical simulation and real data analysis.