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Classes of probability measures built on the properties of Benford’s law

Author

Listed:
  • Roy Cerqueti

    (Sapienza University of Rome
    University of Angers)

  • Mario Maggi

    (University of Pavia
    University of Pavia)

Abstract

Benford’s law is a particular discrete probability distribution that is often satisfied by the significant digits of a dataset. The nonconformity with Benford’s law suggests the possible presence of data manipulation. This paper introduces two novel generalized versions of Benford’s law that are less restrictive than the original Benford’s law—hence, leading to more probable conformity of a given dataset. Such generalizations are grounded on the existing mathematical relations between Benford’s law probability distribution elements. Moreover, one of them leads to a set of probability distributions that is a proper subset of that of the other one. We show that the considered versions of Benford’s law have a geometric representation on the three-dimensional Euclidean space. Through suitable optimization models, we show that all the probability distributions satisfying the more restrictive generalization exhibit at least acceptable conformity with Benford’s law, according to the most popular distance measures. We also present some examples to highlight the practical usefulness of the introduced devices.

Suggested Citation

  • Roy Cerqueti & Mario Maggi, 2025. "Classes of probability measures built on the properties of Benford’s law," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 109(1), pages 197-216, March.
    • Handle: RePEc:spr:alstar:v:109:y:2025:i:1:d:10.1007_s10182-024-00505-2
    DOI: 10.1007/s10182-024-00505-2
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    References listed on IDEAS

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    1. Deckert, Joseph & Myagkov, Mikhail & Ordeshook, Peter C., 2011. "Benford's Law and the Detection of Election Fraud," Political Analysis, Cambridge University Press, vol. 19(3), pages 245-268, July.
    2. repec:bla:germec:v:10:y:2009:i::p:339-351 is not listed on IDEAS
    3. De Ceuster, Marc J. K. & Dhaene, Geert & Schatteman, Tom, 1998. "On the hypothesis of psychological barriers in stock markets and Benford's Law," Journal of Empirical Finance, Elsevier, vol. 5(3), pages 263-279, September.
    4. Alex Ely Kossovsky, 2021. "On the Mistaken Use of the Chi-Square Test in Benford’s Law," Stats, MDPI, vol. 4(2), pages 1-35, May.
    5. Karl‐Heinz Tödter, 2009. "Benford's Law as an Indicator of Fraud in Economics," German Economic Review, Verein für Socialpolitik, vol. 10(3), pages 339-351, August.
    6. Barabesi, Lucio & Pratelli, Luca, 2020. "On the Generalized Benford law," Statistics & Probability Letters, Elsevier, vol. 160(C).
    7. Ausloos, Marcel & Cerqueti, Roy & Mir, Tariq A., 2017. "Data science for assessing possible tax income manipulation: The case of Italy," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 238-256.
    8. Riccioni, Jessica & Cerqueti, Roy, 2018. "Regular paths in financial markets: Investigating the Benford's law," Chaos, Solitons & Fractals, Elsevier, vol. 107(C), pages 186-194.
    9. Pietronero, L. & Tosatti, E. & Tosatti, V. & Vespignani, A., 2001. "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 297-304.
    10. Cerqueti, Roy & Maggi, Mario, 2021. "Data validity and statistical conformity with Benford’s Law," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    11. Nye John & Moul Charles, 2007. "The Political Economy of Numbers: On the Application of Benford's Law to International Macroeconomic Statistics," The B.E. Journal of Macroeconomics, De Gruyter, vol. 7(1), pages 1-14, July.
    12. Ausloos, M. & Herteliu, C. & Ileanu, B., 2015. "Breakdown of Benford’s law for birth data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 736-745.
    13. Mir, T.A., 2014. "The Benford law behavior of the religious activity data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 408(C), pages 1-9.
    14. Tariq Ahmad Mir & Marcel Ausloos & Roy Cerqueti, 2014. "Benford's law predicted digit distribution of aggregated income taxes: the surprising conformity of Italian cities and regions," Papers 1410.2890, arXiv.org.
    15. Ausloos, Marcel & Cerqueti, Roy & Lupi, Claudio, 2017. "Long-range properties and data validity for hydrogeological time series: The case of the Paglia river," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 470(C), pages 39-50.
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