基于单元格可分离动态规划的二维列联表卡方族统计量的精确条件分布
Exact Conditional Distributions of Chi-Square-Family Statistics for Two-Way Contingency Tables, by Cell-Separable Dynamic Programming
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中文总结 AI 辅助
该研究提出基于单元格可分离动态规划的方法，可计算二维列联表卡方族统计量的精确条件分布，为评估近似参照提供无蒙特卡洛的基准真值，适用于稀疏或异质表场景。
中文摘要 AI 辅助
二维列联表的所有卡方族统计量（皮尔逊X²、幂散度成员、方差稳定化T_root）均参照近似零分布（卡方、矩匹配卡方、鞍点或自助法），但该近似在稀疏或异质表上会出现校准误差，而精确分布是原子的格点而非连续曲线。精确分布以两个边际为条件（多元费舍尔非中心超几何分布），但通常被视为不可计算，因为共享同一边际的表数量极为庞大。本文证明，任何卡方族统计量的精确条件矩和精确条件分布均可在不枚举表的情况下计算，原因在于该统计量是单元格可加的，且边际条件律按单元格分解：动态规划每次遍历一个单元格，在每行容量状态下存储少量矩累加器、统计量值到概率的映射或一个复数（特征函数），其计算成本由边际状态数而非表数量决定。矩引擎计算了5×5表（每个边际纤维约790亿个表）的精确条件矩，耗时3秒；分布引擎返回精确尾概率，经枚举验证达到机器精度，而三阶矩卡方拟合的误差高达0.28。该引擎提供了无蒙特卡洛的基准真值，可用于评估任何近似参照，还可扩展到非零备择假设，是配套论文中精确条件区间和报告工具的计算基础。
英文摘要
Every chi-square-family statistic for a two-way contingency table (Pearson's X^2, the power-divergence members, the variance-stabilized T_root) is referred to an approximate null distribution (chi-square, moment-matched chi-square, saddlepoint, or bootstrap) that miscalibrates on sparse or heterogeneous tables, where the exact distribution is a lattice of atoms rather than a continuous curve. The exact reference conditions on both margins (the multivariate Fisher noncentral hypergeometric law) but is usually treated as uncomputable, because the number of tables sharing a margin is astronomical. We show that the exact conditional moments and the exact conditional distribution of any chi-square-family statistic are computable without enumerating tables, because the statistic is additive over cells and the margin-conditional law factorizes cell by cell: a dynamic program walks the table one cell at a time, carrying per row-capacity state either a few moment accumulators, a map from statistic value to probability, or one complex number (the characteristic function), at a cost set by the number of margin states rather than the number of tables. The moment engine computes the exact conditional moments of a 5x5 table at three per cell (about 79 billion tables on the margin fibre) in three seconds; the distribution engine returns the exact tail, verified against enumeration to machine precision, where a three-moment chi-square fit misses it by up to 0.28. The engine furnishes Monte-Carlo-free ground truth against which any approximate reference can be measured, extends to the non-null alternative, and is the computational substrate beneath the exact conditional interval and reporting tools of the companion papers.
