Testing symmetry for additive distortion measurement errors data

This paper studies how to estimate and test the symmetry of a continuous variable under the additive distortion measurement errors setting. The unobservable variable is distorted in a additive fashion by an observed confounding variable. Firstly, a direct plug-in estimation procedure is proposed to calibrate the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are investigated. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.