Empirical likelihood-based inference in linear errors-in-covariables models with validation data

Linear errors-in-covariables models are considered, assuming the availability of independent validation data on the covariables in addition to primary data on the response variable and surrogate covariables.We first develop an estimated empirical loglikelihood with the help of validation data and prove that its asymptotic distribution is that of a weighted sum of independent standard x21 random variables with unknown weights. By estimating the unknown weights consistently, we construct an estimated empirical likelihood confidence region for the regression parameter vector. We also suggest an adjusted empirical loglikelihood and prove that its asymptotic distribution is a standard x2. To avoid estimating the unknown weights or the adjustment factor, we propose a partially smoothed bootstrap empirical loglikelihood for constructing a confidence region which has asymptotically correct coverage probability. A simulation study is conducted to compare the proposed methods with a method based on a normal approximation in terms of coverage accuracy and average length of the confidence interval.