A Finite-sample bias correction method for general linear model in the presence of differential measurement errors

This paper focuses on the general linear measurement error model, in which some or all predictors are measured with error, while others are measured precisely. We propose a semi-parametric estimator that works under general mechanisms of measurement error, including differential and non-differential errors. Other popular methods, such as the corrected score and conditional score methods, only work for non-differential measurement error models, but our estimator works in all scenarios. We develop our estimator by considering a family of objective functions that depend on an unspecified weight function. Using statistical error analysis and perturbation theory, we derive the optimal weight function under the small-sigma regime. The resulting estimator is statistically optimal in all senses. Even though we develop it under the small-sigma regime, we also establish its consistency and asymptotic normality under the large sample regime. Finally, we conduct a series of numerical experiments to confirm that the proposed estimator outperforms other existing methods.