BET on Independence

We study the problem of nonparametric dependence detection. Many existing methods may suffer severe power loss due to nonuniform consistency, which we illustrate with a paradox. To avoid such power loss, we approach the nonparametric test of independence through the new framework of binary expansion statistics (BEStat) and binary expansion testing (BET), which examine dependence through a novel binary expansion filtration approximation of the copula. Through a Hadamard transform, we find that the symmetry statistics in the filtration are complete sufficient statistics for dependence. These statistics are also uncorrelated under the null. By using symmetry statistics, the BET avoids the problem of nonuniform consistency and improves upon a wide class of commonly used methods (a) by achieving the minimax rate in sample size requirement for reliable power and (b) by providing clear interpretations of global relationships upon rejection of independence. The binary expansion approach also connects the symmetry statistics with the current computing system to facilitate efficient bitwise implementation. We illustrate the BET with a study of the distribution of stars in the night sky and with an exploratory data analysis of the TCGA breast cancer data. Supplementary materials for this article are available online.本文研究非参数依赖检测问题。现有诸多方法因非一致相合性可能遭遇严重的检验功效损失，本文通过一则悖论对此予以阐释。为规避此类功效损失，本文借助二进制展开统计量（Binary Expansion Statistics, BEStat）与二进制展开检验（Binary Expansion Testing, BET）这一新框架开展独立性非参数检验，该框架通过一种新颖的连接函数（Copula）二进制展开滤过近似来刻画变量间的依赖性。通过阿达马变换（Hadamard Transform），本文发现该滤过中的对称统计量是刻画依赖性的完备充分统计量，且在原假设下，此类统计量彼此互不相关。通过采用对称统计量，BET规避了非一致相合性问题，并在一类广泛使用的方法上实现了性能提升：其一，在满足可靠检验功效的样本量要求上达到了极小极大率；其二，在拒绝独立性假设时，可对全局关联关系给出清晰的解释。二进制展开方法还将对称统计量与现有计算系统相衔接，便于实现高效的按位运算。本文通过夜空恒星分布研究与癌症基因组图谱（The Cancer Genome Atlas, TCGA）乳腺癌数据集的探索性数据分析，对BET方法进行了实例验证。本文的补充材料可在线获取。