顺序统计量的度量：信息论视角
Measuring Dependencies of Order Statistics: An Information Theoretic Perspective
摘要：
考虑从某个已知的抽样分布$P_X$独立且同分布地抽取的随机样本$X_1 , X_2 , ..., X_n$。令$X_{(1)} \le X_{(2)} \le ... \le X_{(n)}$表示样本的顺序统计量。本文的第一部分主要讨论具有可逆累积分布函数的分布。在此假设下，建立了一个与分布无关的性质，表明顺序统计量的联合分布与顺序统计量的乘积分布之间的$f$ -散度不依赖于原始抽样分布$P_X$ .此外，它示出的顺序统计量的两个子集之间的互信息也满足分布的自由属性，即它不依赖于$P_X$。此外，$X_{(r)}$和$X_{(m)}$之间的去耦率（即，互信息接近零的速率）被表征用于$(r,m)$的各种选择。本文的第二部分考虑了一个离散分布族，它不满足本文第一部分中的假设。与第一部分的结果相比，它表明，在离散设置，顺序统计量之间的互信息依赖于抽样分布$P_X$。尽管如此，第一部分的结果仍然可以用作解耦率的上限。
英文摘要：
Consider a random sample $X_1 , X_2 , ..., X_n$ drawn independently and identically distributed from some known sampling distribution $P_X$. Let $X_{(1)} \le X_{(2)} \le ... \le X_{(n)}$ represent the order statistics of the sample. The first part of the paper focuses on distributions with an invertible cumulative distribution function. Under this assumption, a distribution-free property is established, which shows that the $f$-divergence between the joint distribution of order statistics and the product distribution of order statistics does not depend on the original sampling distribution $P_X$. Moreover, it is shown that the mutual information between two subsets of order statistics also satisfies a distribution-free property; that is, it does not depend on $P_X$. Furthermore, the decoupling rates between $X_{(r)}$ and $X_{(m)}$ (i.e., rates at which the mutual information approaches zero) are characterized for various choices of $(r,m)$. The second part of the paper considers a family of discrete distributions, which does not satisfy the assumptions in the first part of the paper. In comparison to the results of the first part, it is shown that in the discrete setting, the mutual information between order statistics does depend on the sampling distribution $P_X$. Nonetheless, it is shown that the results of the first part can still be used as upper bounds on the decoupling rates.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
