Jackknife empirical likelihood for the correlation coefficient with multiplicative distortion measurement errors

In this paper, we consider the estimation problem of a correlation coefficient between two unobserved variables of interest that are distorted in a multiplicative way by some unobserved confounding variable. We investigate the direct plug-in estimator of the correlation coefficient. We propose using jackknife empirical likelihood (JEL) and its variations to construct confidence intervals for the correlation coefficient based on the estimator. The proposed JEL statistic is shown to be asymptotically a standard chi-squared distribution. We compare our methods to the previous empirical likelihood (EL) techniques of Zhang et al. (2014, 'A Revisit to Correlation Analysis for Distortion Measurement Error Data', Journal of Multivariate Analysis, 124, 116–129) and show the JEL possesses better small sample properties. Simulation studies are conducted to examine the performance of the proposed estimator, and we also use our proposed methods to analyse the Boston housing data for illustration.