对系统树平衡敏感的两个统计量的均值、方差和极限分布
The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance
摘要：
二十年来，Colless指数一直是评估系统发育树平衡性最常用的统计方法。在这篇文章中，这一统计研究的Yule和统一模型的系统发育树。分析的主要工具是耦合参数与另一个著名的指数称为萨金统计。这两个统计量的平均值，方差和协方差的渐近，以及它们的极限联合分布的大概率。在Yule模型下，极限分布作为泛函不动点方程的解而出现。在均匀模型下，极限分布是Airy分布。本研究的基石是这样一个事实，即系统发生树的概率模型与二叉搜索树的随机排列和加泰罗尼亚模型密切相关。
英文摘要：
For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.
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