Chilean Journal of Statistics 17(1), pp. 68–87.
ISSN/ISBN: Not available at this time. DOI: 10.32372/ChJS.17-01-04
Abstract: Goodness-of-fit tests for the Benford law can falsely flag authentic data when their first digits arise from an inverse power law. In this article, we show that, for any finite power-law exponent, the chi-squared test rejects such data with probability tending to one as the sample size increases, which establishes a model-misspecification problem. To address this problem, we develop a corrected method based on the power-law first- digit statistical distribution. In addition, our method jointly estimates the power-law exponent and scale phase by using the minimum chi-squared technique and measures the remaining deviation from the fitted first-digit baseline. Under regularity conditions, the associated corrected statistic follows an asymptotic chi-squared distribution with six degrees of freedom. Furthermore, a bootstrap equivalence decision is used to evaluate compatibility with the fitted power-law first-digit family. In reproducible simulations and public datasets, the corrected method distinguishes data compatible with a unit-scale power-law first-digit distribution from the selected fabricated, retracted, and bounded- support benchmarks. The developed method is intended for quantities spanning several orders of magnitude.
Bibtex:
@article{,
author = {Francisco Bustillos and Victor Leiva},
title = {When the Benford law might fail: A corrected first-digit test for power-law distributions},
year = {2026},
journal = {Chilean Journal of Statistics},
volume = {17},
number = {1},
pages = {68—87},
doi = {10.32372/ChJS.17-01-04},
url = {https://soche.cl/chjs/volumes/17/ChJS-17-01-04.pdf},
}
Reference Type: Journal Article
Subject Area(s): Statistics