Exponential calibration for correlation coefficient with additive distortion measurement errors

This paper studies the estimation of correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in an additive fashion by an observed confounding variable. We propose a new identifiability condition by using the exponential calibration to obtain calibrated variables and propose a direct‐plug‐in estimator for the correlation coefficient. We show that the direct‐plug‐in estimator is asymptotically efficient. Next, we suggest an asymptotic normal approximation and an empirical likelihood‐based statistic to construct the confidence intervals. Last, we propose several test statistics for testing whether the true correlation coefficient is zero or not. The asymptotic properties of the proposed 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 temperature forecast data set for an illustration.