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.

中文翻译：

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张和郑的“具有指数衰减偏差的互信息估计器”

Zhang, Z. 和 Zheng, L. (2015)：“具有指数衰减偏差的互信息估计器”，Stat。应用程序。热内。摩尔。Biol., 14, 243–252，提出了一种在熵视角下开发的互信息非参数估计器，并证明它比插件估计器具有更小的偏差，但在某些条件下具有相同的渐近正态性。然而，在他们的文章中错误地建议渐近正态性可用于测试联合字母表上两个随机元素之间的独立性。当两个随机元素独立时，其渐近分布 $\sqrt{n}$ n -normed 估计量退化，因此声称的正态性不成立。本文通过建立一个新的卡方检验来补充张和郑，该检验使用相同的熵统计量，互信息为零。Zhang 和 Zheng 中的三个例子是使用新的测试重新处理的。结果证明更加明智，并进一步说明了熵视角在字母统计推断中的优势。更具体地说，在示例 2 中，当已知存在正互信息时，新测试会检测到它，但对数似然比测试无法检测到它。