This paper considers linear regression models when neither the response variable nor the covariates can be directly observed, but are measured with multiplicative distortion measurement errors. To eliminate the effect caused by the distortion, we propose two calibration procedures: the conditional absolute mean calibration and the conditional variance calibration. Both calibration procedures avoid using the nonzero expectation conditions imposed on the variables in the literature. Utilizing these calibrated variables, the least squares estimators are obtained, associated with their asymptotic results. The asymptotic normal confidence intervals and empirical likelihood confidence intervals are also proposed. Simulation studies are conducted to compare the proposed calibration procedures and a real example is analyzed to illustrate our proposed method.
"Calibration procedures for linear regression models with multiplicative distortion measurement errors." Braz. J. Probab. Stat. 34 (3) 519 - 536, August 2020. https://doi.org/10.1214/19-BJPS451