Benford检验实际检验的是什么？边际符合性、抽样结构与法证推断

中文摘要 AI 辅助

Benford定律规定了有效数字的边际分布，而通常的首位数字Pearson $p$-值是在独立多项抽样模型下校准的。我们通过四种构造来区分这两种陈述，这些构造共享相同的一次性或合并Benford目标，但具有不同的联合结构。在包裹高斯圆形马尔可夫构造下，名义5%的Pearson检验在固定持续性水平下拒绝了21.6%的样本，尽管每个一次性边际分布恰好是Benford的；序列级二阶矩匹配校准将拒绝率降至4.8%。同样的策略在随机旋转和随机组合设计中表现良好。该效应在样本量和序列长度的变化中持续存在，并且对连续对数有效数字Cramér-von Mises差异的网格近似显示了相同的设计依赖性。在此处考察的包裹高斯设计中，校正后的程序保持拒绝概率随平滑边际偏离的大小而上升。最后，我们将校准与信息边界区分开来：仅有效数字方法无法检测到不改变完整有效数字过程的变更，但它们能轻易检测到改变该过程的扰动。Benford符合性是一个边际陈述；Benford $p$-值仅相对于指定的统计量、抽样定律和校准程序才有效；完整性声明需要实质性的竞争模型。

英文摘要

Benford's law specifies a marginal distribution for significant digits, whereas the usual first-digit Pearson $p$-value is calibrated under an independent multinomial sampling model. We separate these statements with four constructions that share the same one-time or pooled Benford target but have different joint structures. Under a wrapped-Gaussian circular Markov construction, the nominal 5% Pearson test rejects 21.6% of samples at a fixed persistence level even though every one-time marginal is exactly Benford; a sequence-level second-order moment-matching calibration reduces the rejection rate to 4.8%. The same strategy performs well in the randomized-rotation and random-composition designs. The effect persists across changes in sample size and sequence length, and a grid approximation to a continuous log-significand Cramér-von Mises discrepancy shows the same design dependence. In the wrapped-Gaussian design examined here, corrected procedures retain rejection probabilities that rise with the size of a smooth marginal departure. Finally, we distinguish calibration from an information boundary: significand-only methods cannot detect changes that leave the complete significand process unchanged, but they readily detect perturbations that alter it. Benford conformity is a marginal statement; a Benford $p$-value is valid only relative to a specified statistic, sampling law, and calibration procedure; an integrity claim requires substantive competing models.

发表机构

• Université du Québec à Montréal (UQAM)（蒙特利尔大学）
• Kyoto University（京都大学）

机构由 AI 辅助整理，请以论文原文为准。