The correlation coefficient serves as a key metric in statistical analysis, however, its estimation may be affected by the bias when confounding variables introduce additive distortion errors. This research explores the reliability of the jackknife empirical likelihood (JEL) approach and its extension, the adjusted JEL (AJEL), in estimating the correlation coefficient across various distortion settings. To address the challenges posed by complex distortions with exponential and periodic components, we propose a log-adjusted residual estimator, which stabilizes the variance and reduces the bias by applying a log-transformation to the conditional expectation of the observed data.

We evaluate this estimator alongside the standard approach, considering three types of distortion functions: linear, quadratic, and periodic, as well as three distributions for the confounding variable: uniform, normal, and beta. Through theoretical derivations and simulation studies, we demonstrate that the log-adjusted residual estimator enhances the performance of JEL-based methods, particularly for the quadratic and periodic distortions, improving coverage probabilities and confidence interval lengths.

Furthermore, our analysis of JEL and its variant across various distortion scenarios confirms their robustness. These findings highlight the adaptability of JEL-based methods to complex distortion structures, with the proposed estimator offering a reliable advancement for practical applications.