Generalised Benford Law
Benford's law [1] crops up in the most unexpected places. If you are not familiar with it, an informal place to get started may be Netflix's documentary "Digits" from the series "Connected: The Hidden Science of Everything".
Why does Benford's law happen at all? Is there a more general law behind it? Using scale-invariance considerations, Pietronero et al. first discovered that a more general law is possible [2], and Barabesi and Pratelli then extended their work [3]. In our paper [4] we have in turn rediscovered and further extended these generalised results, and even uncovered a previously unknown result for the classic Benford distribution: a closed-form expression for the distribution of the j-th most significant digit (for j>=2).
The quintessential application of Benford's law is forensic analysis. In [4], we also question the optimality of using most significant digits in forensic detection tests: we show that every time that Benford's "law" holds for a dataset, an alternative "law" for the continued fraction coefficients of the logarithm of the same data also holds. That is to say, there is an analogous of Benford's law for continued fraction coefficients.
In [5] we provide general models for most significant digits and leading continued fraction coefficients. The latter can provide alternative forensic tests, equivalent to using most significant digits.
See the presentation that we gave in EUSIPCO 2021. You can also download a Matlab toolbox reproducing all the results in [5] and [4]. Enjoy!
Félix Balado and Guénolé Silvestre
References
[1] F. Benford, "The law of anomalous numbers", Proceedings of the American Philosophical Society, 78(4):551--572, 1938.
[2] L. Pietronero, E. Tosatti, V. Tosatti, and A. Vespignani, "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf", Physica A: Statistical Mechanics and its Applications, 293(1):297--304, 2001.
[3] L. Barabesi and L. Pratelli, "On the generalized Benford law", Statistics & Probability Letters, 160:108702, 2020.
[4] F. Balado and G. Silvestre, "Benford's Law: Hammering a Square Peg Into a Round Hole?", 29th European Conference on Signal Processing (EUSIPCO), Dublin, Ireland, August, 2021, pp. 796--800. [full text]
[5] F. Balado and G. Silvestre, "General Distributions of Number Representation Elements", arXiv:2301.10547 To appear in Probability in the Engineering and Informational Sciences [full text] (Cambridge University Press), 2024.
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