Statistical models and the Benford hypothesis: a unified framework

Lucio Barabesi (), Andrea Cerioli () and Marco Marzio ()
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Lucio Barabesi: University of Siena
Andrea Cerioli: University of Parma
Marco Marzio: “G. D’Annunzio” University

TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, 2023, vol. 32, issue 4, No 17, 1479-1507

Abstract: Abstract The Benford hypothesis is the statement that a random sample is made of realizations of an absolutely continuous random variable distributed according to Benford’s law. Its potential interest spans over many domains such as detection of financial frauds, verification of electoral processes and investigation of scientific measurements. Our aim is to provide a principled framework for the statistical evaluation of this statement. First, we study the probabilistic structure of many classical univariate models when they are framed in the space of the significand and we measure the closeness of each model to the Benford hypothesis. We then obtain two asymptotically equivalent and powerful tests. We show that the proposed test statistics are invariant under scale transformation of the data, a crucial requirement when compliance to the Benford hypothesis is used to corroborate scientific theories. The empirical advantage of the proposed tests is shown through an extensive simulation study. Applications to astrophysical and hydrological data also motivate the methodology.

Keywords: Benford’s law; Digit distribution; Goodness-of-fit; Kolmogorov distance; Likelihood ratio test; Significand; 62E10; 62F03; 62G07; 65C05 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11749-023-00881-y

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