Empirical likelihood inference in nonlinear errors-in-covariables models with validation data

In this article we study inference in parametric-nonparametric errors-in-covariables regression models using an empirical likelihood approach based on validation data. It is shown that the asymptotic behavior of the proposed estimator depends on the ratio of the sizes of the primary sample and the validation sample, respectively. Unlike cases without measurement errors, the limit distribution of the estimator is no longer tractable and cannot be used for constructing confidence regions. Monte Carlo approximations are employed to simulate the limit distribution. To increase the coverage accuracy of confidence regions, two adjusted empirical likelihood estimators are recommended, which in the limit have a standard chi-squared distribution. A simulation study is carried out to compare the proposed methods with other existing methods. The new methods outperform the least squares method, and one of them works better than simulation-extrapolation (SIMEX) estimation, even when the restrictive model assumptions needed for SIMEX are satisfied. An application to a real dataset illustrates our new approach.