论文标题
测量订单统计的依赖性：信息理论观点
Measuring Dependencies of Order Statistics: An Information Theoretic Perspective
论文作者
论文摘要
考虑一个随机样本$ x_1，x_2，...，x_n $独立绘制，并从某些已知的采样分发$ p_x $分布。令$ x _ {（1）} \ le x _ {（2）} \ le ... \ le x _ {（n）} $表示样本的顺序统计信息。本文的第一部分着重于具有可逆累积分布函数的分布。在此假设下，建立了无分配的财产，这表明订单统计的联合分布与订单统计的产品分布之间的$ f $ divergence不取决于原始采样分布$ p_x $。此外，结果表明，订单统计的两个子集之间的共同信息也满足了无分配的财产。也就是说，它不取决于$ p_x $。此外，$ x _ {（r）} $和$ x _ {（m）} $之间的去耦率（即，相互信息接近零）的速率是$（r，m）$的各种选择的特征。本文的第二部分考虑了一个离散分布的家族，该家族不满足本文第一部分中的假设。与第一部分的结果相比，显示在离散设置中，订单统计信息之间的共同信息确实取决于采样分布$ p_x $。但是，结果表明，第一部分的结果仍然可以用作脱钩速率上的上限。
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.
