On “A mutual information estimator with exponentially decaying bias” by Zhang and Zheng

Zhang, Z. and Zheng, L. (2015): “A mutual information estimator with exponentially decaying bias,” Stat. Appl. Genet. Mol. Biol., 14, 243–252, proposed a nonparametric estimator of mutual information developed in entropic perspective, and demonstrated that it has much smaller bias than the plugin estimator yet with the same asymptotic normality under certain conditions. However it is incorrectly suggested in their article that the asymptotic normality could be used for testing independence between two random elements on a joint alphabet. When two random elements are independent, the asymptotic distribution of $\sqrt{n}$n-normed estimator degenerates and therefore the claimed normality does not hold. This article complements Zhang and Zheng by establishing a new chi-square test using the same entropic statistics for mutual information being zero. The three examples in Zhang and Zheng are re-worked using the new test. The results turn out to be much more sensible and further illustrate the advantage of the entropic perspective in statistical inference on alphabets. More specifically in Example 2, when a positive mutual information is known to exist, the new test detects it but the log likelihood ratio test fails to do so.