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 log-likelihood with the help of validation data and prove that its asymptotic distribution is that of a weighted sum of independent standard x random variables with unknown weights. By estimating the unknown weights consistently, an estimated empirical likelihood confidence region on the regression parameter vector is constructed. We also suggest an adjusted empirical log-likelihood and prove that its asymptotic distribution is a standard X To avoid estimating the unknown weights or the adjustment factor, we propose a partially smoothed bootstrap empirical log- likelihood to construct a confidence region which has asymptotically correct coverage probability. A simulation study is conducted to compare the proposed methods with a normal approximation based method in term of coverage accuracy and average length of the confidence interval.