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arXiv · 2609.18424

What Does a Benford Test Actually Test? Marginal Conformity, Sampling Structure, and Forensic Inference

Arthur Charpentier
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Abstract

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.

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Arthur Charpentier. 2026-09-16. What Does a Benford Test Actually Test? Marginal Conformity, Sampling Structure, and Forensic Inference. https://arxiv.org/abs/2609.18424

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