Covariance ratio under multiplicative distortion measurement errors

We propose a covariance ratio measure for symmetry or asymmetry of a probability density function. This measure is constructed by the ratio of the covariance connected with the density function and the distribution function. We first propose a non parametric moment-based estimator of the covariance ratio measure and study its asymptotic results. Next, we consider statistical inference of the covariance ratio measure by using the empirical likelihood method. The empirical likelihood statistic is shown to be asymptotically a standard chi-squared distribution. Last, we study the covariance ratio measure when the random variable is unobserved under the multiplicative distortion measurement errors setting. The density function and the distribution function of the unobserved variable are estimated by using four calibrated variables. Appealing to these estimators, the calibrated covariance ratio measures are proposed and further shown to be asymptotically efficient as if there are no multiplicative distortion effects. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test procedures. These methods are applied to analyze two real datasets for illustration.