发表机构
University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对双向表独立性的稀疏偏离问题，提出含异方差轮廓与检测边界的理论，给出自适应高阶批评门限及精确锚点检验，验证了5543个表格语料库中三分之二采用精确锚点的结论。
AI 中文摘要
检验R×C表的独立性通常采用一个综合统计量对抗密集备择假设；当偏离是稀疏的、集中在少数单元格时，核心对象则是检测边界，即任何检验都能发现的最弱信号。本文从该视角处理独立性问题，分为三个部分：第一部分将稀疏信号检测理论（Ingster 1997；Donoho和Jin 2004；Chhor、Mukherjee和Sen 2024）专门应用于表格，将其视为异方差高斯序列，其方差轮廓是期望计数表，通过乘积CV恒等式CV_e²=(1+CV_r²)(1+CV_c²)-1由两个边缘固定，边缘异质性恰好是异方差轮廓；方差稳定根变换给出精确标准化幅度2√m(√(1+a)-1)和闭式分离半径，该结果在计数增长条件下有效，且在固定小计数时必然失效。第二部分将独立性视为对数线性无交互模型的拟合优度，证明具有闭式Jaeschke-Eicker零分布的高阶批评（higher criticism）可在未知稀疏性上自适应达到该边界，同时明确在极稀疏强 regime中最大单元格规则占优，且精确条件最优性是推测性的。第三部分测量小计数校准失效（即使在均匀边缘下，大小为0.3至0.6，驱动因素是单个单元格尾部而非异质性），通过精确边缘条件参考定律消除该失效，证明对检验而言，对两个边缘进行条件化是完整处理方式（无需先验；大小为0.003至0.009）；基于边缘轮廓的路由规则将大计数表发送至渐近门限，小计数表发送至精确锚点。5543个公开表格语料库中的三分之二路由至精确锚点，所有结论均通过公开存储的确定性种子代码数值验证。
英文摘要
Two-way contingency tables are tested for independence throughout applied statistics (genomics, network and text co-occurrence, pharmacovigilance, ecology, survey cross-tabulation), and the routine test reads asymptotic Gaussian tail probabilities off the table cell by cell. On the tables people actually analyze this is badly miscalibrated: across 5,543 real public two-way tables, two-thirds have counts small enough or margins heterogeneous enough that the asymptotic maximum-cell independence scan false-positives at a mean 33% against a 0.05 target, while an exact margin-conditional anchor holds at about 1%. The failure is sharpest when the departure is sparse, the association concentrated in a few cells, where the omnibus chi-square is inefficient and the sharp object is the detection boundary. In three parts we answer where a sparse signal can be seen, which combiner attains that limit, and how to calibrate at small counts. Part I specializes the sparse-detection theory of Ingster (1997), Donoho and Jin (2004), and Chhor, Mukherjee and Sen (2024) to independence: the table is a heteroscedastic Gaussian sequence whose variance profile is the expected-count table, fixed by the margins via CVe^2 = (1 + CVr^2)(1 + CVc^2) - 1, giving an exact signal map and a closed-form separation radius valid under a moderate-deviation growing-count condition. Part II shows higher criticism, with its closed-form Jaeschke-Eicker null, attains that boundary adaptively over unknown sparsity. Part III measures the small-count calibration failure (size 0.3 to 0.6 even under uniform margins, so the driver is the per-cell tail), removes it with exact margin-conditional laws (size 0.003 to 0.009), and gives a routing rule sending large-count tables to the asymptotic gate and small-count tables to the exact anchor. Every claim is reproduced from openly deposited, deterministically seeded code.
Comments29 pages, 11 figures, 4 tables. Reproducibility package: doi:10.5281/zenodo.21844797
