A revisit to correlation analysis for distortion measurement error data

In this paper, we consider the estimation problem of a correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a multiplicative fashion by an observed confounding variable. Two estimators, the moment-based estimator and the direct plug-in estimator, are proposed, and we show their asymptotic normality. Moreover, the direct plug-in estimator is shown asymptotically efficient. Furthermore, we suggest a bootstrap procedure and an empirical likelihood-based statistic to construct the confidence interval. The empirical likelihood statistic is shown to be asymptotically chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators. These methods are applied to analyze the Boston housing price data as an illustration.