标题： 列联表的互信息及相关不等式 显示英文标题

标题： Mutual information of Contingency Tables and Related Inequalities

摘要： 为了检验独立性，非常流行使用 $\chi^{2}$-统计量或者 $G^{2}$-统计量（互信息）。渐近地，两者都服从 $\chi^{2}$-分布，因此一个显而易见的问题是哪一种统计量的分布最接近于 $\chi^{2}$-分布。 令人惊讶的是，互信息的分布比 $\chi^{2}$-统计量更接近于 $\chi^{2}$-分布。 出于技术原因，我们将集中于最简单的情形，即自由度为一的情况。 我们引入符号化的对数似然，并证明其分布函数可以由标准正态分布函数通过不等式来关联。 对于超几何分布，我们提出了一个关于符号化对数似然与标准正态分布接近程度的一般猜想，这个猜想比之前发表的结果提供了更为精确的此类分布尾部概率估计。 该猜想已在检验独立性相关的所有相关情形下被数值验证，并且给出了更多其有效性的证据。 显示英文摘要

摘要： For testing independence it is very popular to use either the $\chi^{2}$-statistic or $G^{2}$-statistics (mutual information). Asymptotically both are $\chi^{2}$-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the $\chi^{2}$-distribution. Surprisingly the distribution of mutual information is much better approximated by a $\chi^{2}$-distribution than the $\chi^{2}$-statistic. For technical reasons we shall focus on the simplest case with one degree of freedom. We introduce the signed log-likelihood and demonstrate that its distribution function can be related to the distribution function of a standard Gaussian by inequalities. For the hypergeometric distribution we formulate a general conjecture about how close the signed log-likelihood is to a standard Gaussian, and this conjecture gives much more accurate estimates of the tail probabilities of this type of distribution than previously published results. The conjecture has been proved numerically in all cases relevant for testing independence and further evidence of its validity is given.