Inference on data with both multiplicative and additive measurement errors

Measurement errors are omnipresent in many fields and can lead to serious problems in statistical analysis. In the literature, measurement errors are often assumed to be either additive or multiplicative. We consider the case where a variable is subject to both additive and multiplicative errors. We establish the identifiability and propose a moment-based estimator for the variances of these two types of errors, which is shown to be consistent. We further derive the asymptotic distribution of the estimator and conduct hypothesis tests to examine the existence of the two types of errors. We also develop a likelihood-based method to approximate the density of the error-prone variable. We apply our strategy in the context of linear regression and study its effect on the estimation of regression parameters in combination with Regression Calibration and Simulation Extrapolation. The proposed methodology is numerically investigated through simulations and is implemented in a real data application.