Symmetry measures under additive distortion measurement errors

We examine the symmetry or asymmetry of the density function of a continuous variable under the setting of additive distortion measurement errors. The unobservable variable is distorted in an additive manner by an observed confounding variable. We adopt the covariance ratio-based measure under conditional mean calibration and conditional exponential mean calibration methods. Using the empirical likelihood method, we consider statistical inference of the covariance ratio-based measure under additive distortions to check for symmetry. The calibrated covariance ratio-based measures are shown to be asymptotically efficient as if there were no additive distortion effects. Additionally, Gupta’s skewness with conditional exponential mean calibration is also studied. We conduct Monte Carlo simulation experiments to access the performance of the calibrated measures and test procedures. These methods are applied to analyze a real dataset for illustrative purposes.