This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of 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 examined. 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.
"Measuring symmetry and asymmetry of multiplicative distortion measurement errors data." Braz. J. Probab. Stat. 34 (2) 370 - 393, May 2020. https://doi.org/10.1214/19-BJPS432