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An Alternative to the Oversimplifying Benford’s Law in Experimental Fields

Sankhya B volume 84, pages 778–808 (2022)Cite this article

Abstract

A way to model the distribution of first digits in some naturally occurring collections of data is here highlighted. The proportion of d as leading digit, d ∈⟦1,9⟧, in data is sometimes more likely to follow a specific law whose probability distribution is determined by a lower and an upper bound, rather than Benford’s Law, as one might have expected. These peculiar probability distributions fluctuate around Benford’s values, such fluctuations having often been observed in the literature in experimental data sets (where the physical, biological or economical quantities considered are lower and upper bounded). Knowing beforehand the values of these bounds enables to find, through the developed model, a better adjusted law than Benford’s one.

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References

  • Adresse et géolocalisation des établissements d’enseignement du premier et second degrés (2017). https://www.data.gouv.fr/fr/datasets/adresse-et-geolocalisation-des-etablissements-denseignement-du-premier-et-second-degres-1/https://www.data.gouv.fr/fr/datasets/adresse-et-geolocalisation-des-etablissements-denseignement-du-premier-et-second-degres-1/.

  • Alexopoulos, T. and Leontsinis, S. (2014). Benford’s law in astronomy. J. Astrophys. Astron. 35, 639–648.

    Article  Google Scholar 

  • Alves, A. D., Yanasse, H. H. and Soma, N. Y. (2014). Benford’s law and articles of scientific journals: comparison of JCR and Scopus data. Scientometrics98, 173–184.

    Article  Google Scholar 

  • Aron, J. (2013). Crime-fighting maths law confirms planetary riches. New Scientist 220, 12, 194–205.

    Google Scholar 

  • Arshadi, L. and Jahangir, A. H. (2014). Benford’s law behavior of internet traffic. J. Netw. Comput. Appl. 40, 194–205.

    Article  Google Scholar 

  • Beer, T. W. (2009). Terminal digit preference: beware of Benford’s law. J. Clin. Pathol. 62, 192.

    Article  Google Scholar 

  • Benford, F. (1938). The law of anomalous numbers. Proc. Am. Philos. Soc. 78, 127–131.

    MATH  Google Scholar 

  • Blondeau Da Silva, S. (2019). Benford or not benford: a systematic but not always well-founded use of an elegant law in experimental fields. Commun. Math. Stat. https://doi.org/10.1007/s40304-018-00172-1.

  • Blondeau Da Silva, S. (2020a). BeyondBenford: compare the goodness of fit of Benford’s and Blondeau Da Silva’s digit distributions to a given dataset. R package version 1.4.

  • Blondeau Da Silva, S. (2020b). Limits of Benford’s law in experimental field. Int. J. Appl. Math. 33, 4, 685–695.

    Google Scholar 

  • Blondeau Da Silva, S. (2021). BeyondBenford R package to compare Benford’s and BDS’s distributions. Computer Science Journal of Moldova 29, 241–256.

    Google Scholar 

  • Cella, R. S. and Zanolla, E. (2018). Benford’s law and transparency: an analysis of municipal expenditure. Braz. Bus. Rev. 15, 331–347.

    Article  Google Scholar 

  • Cho, W. K. T. and Gaines, B. J. (2007). Breaking the (Benford) law: statistical fraud detection in campaign finance. Am. Stat. 61, 218–223.

    Article  MathSciNet  Google Scholar 

  • Clippe, P. and Ausloos, M. (2012). Benford’s law and Theil transform of financial data. Phys. A: Stat. Mech. Appl. 391, 6556–6567.

    Article  Google Scholar 

  • Cournane, S., Sheehy, N. and Cooke, J. (2014). The novel application of Benford’s second order analysis for monitoring radiation output in interventional radiology. Phys. Med. 30, 413–418.

    Article  Google Scholar 

  • Deckert, J., Myagkov, M. and Ordeshook, P. (2011). Benford’s law and the detection of election fraud. Polit. Anal. 19, 245–268.

    Article  Google Scholar 

  • Diekmann, A. (2007). Not the first digit! Using Benford’s law to detect fraudulent scientific data. J. Appl. Stat. 34, 321–329.

    Article  MathSciNet  MATH  Google Scholar 

  • Fewster, R. M. (2009). A simple explanation of Benford’s law. Am. Stat. 63, 26–32.

    Article  MathSciNet  Google Scholar 

  • Friar, J.L., Goldman, T. and Pérez-Mercader, J. (2012). Genome sizes and the Benford distribution. Plos One 7(5).

  • Gauvrit, N. and Delahaye, J. -P. (2011). Scatter and regularity imply Benford’s law... and more, pp 53–69. World Scientific.

  • Golbeck, J. (2015). Benford’s law applies to online social networks. Plos One 10(8).

  • Goodman, W. (2016). The promises and pitfalls of Benford’s law. Significance 13, 38–41.

    Article  Google Scholar 

  • Hickman, M. J. and Rice, S. K. (2010). Digital analysis of crime statistics: does crime conform to Benford’s law? J. Quant. Criminol. 26, 333–349.

    Article  Google Scholar 

  • Hill, T. P. (1988). Random-number guessing and the first digit phenomenon. Psychol. Rep. 62, 967–971.

    Article  Google Scholar 

  • Janvresse, E. and De La Rue, T. (2004). From uniform distributions to Benford’s law. J. Appl. Probab. 41, 1203–1210.

    Article  MathSciNet  MATH  Google Scholar 

  • Kreuzer, M., Jordan, D., Antkowiak, B., Drexler, B., Kochs, E. F. and Schneider, G. (2014). Brain electrical activity obeys Benford’s law. Anesth. Analg. 118, 183–191.

    Article  Google Scholar 

  • Lacasa, L. and Fernandez-Gracia, J. (2019). Election forensics: quantitative methods for electoral fraud detection. Forensic Sci. Int. 294, e19–e22.

    Article  Google Scholar 

  • Mebane, W. R. . (2006). Election forensics: the second-digit Benford’s law test and recent American presidential elections. In Proceedings of the election fraud conference.

  • Mir, T. A. (2014). The Benford law behavior of the religious activity data. Phys. A: Stat. Mech. Appl. 408, 1–9.

    Article  Google Scholar 

  • Newcomb, R. (1881). Note on the frequency of use of the different digits in natural numbers. Am. J. Math. 4, 39–40.

    Article  MathSciNet  MATH  Google Scholar 

  • Nigrini, M. (2012). Benford’s law: applications for forensic accounting, auditing, and fraud detection, vol. 586. Wiley.

  • Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philos. Mag. 50, 157–175.

    Article  MATH  Google Scholar 

  • Raimi, R. A. (1976). The first digit problem. Am. Math. Mon. 83, 521–538.

    Article  MathSciNet  MATH  Google Scholar 

  • Rauch, B., Göttsche, M., Brälher, G. and Engel, S. (2011). Fact and fiction in EU-governmental economic data. Ger. Econ. Rev. 12, 243–255.

    Article  Google Scholar 

  • Sambridge, M., Tkalcic, H. and Arroucau, P. (2011). Benford’s law of first digits: from mathematical curiosity to change detector. Asia Pac. Math. Newsl. 1, 1–6.

    MathSciNet  Google Scholar 

  • Saville, A. (2006). Using Benford’s law to detect data error and fraud: an examination of companies listed on the Johannesburg Stock Exchange. South Afr. J. Econ. Manag. Sci. 9, 341–354.

    Article  Google Scholar 

  • Scott, P. D. and Fasli, M (2001). Benford’s law: an empirical investigation and a novel explanation. CSM Technical Report 349 University of Essex. https://cswww.essex.ac.uk/technical-reports/2001/CSM-349.pdf.

  • Service d’Administration Nationale des Données et Référentiels sur l’Eau (2017). Cours d’eau - Métropole 2017 - BD Carthage http://wnce.fr/atlas/srv/fre/catalog.search#/metadata/7381de46-42f7-42df-9abe-0ecd4b946034.

  • Smith, S. W. (2012). The scientist and engineer’s guide to digital signal processing. Chapter 34: Explaining Benford’s Law.

  • Tolle, C., Budzien, J., and Laviolette, R., (2000). Do dynamical systems follow Benford’s law? Chaos 10(2).

  • Van Rossum, G. (1995). Python tutorial, volume Technical Report CS-R9526. Centrum voor Wiskunde en Informatica (CWI).

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  1. Rectorat de Limoges, 13, rue Francois Chenieux, 87000, Limoges, France

    Stéphane Blondeau Da Silva

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  1. Stéphane Blondeau Da Silva
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Correspondence to Stéphane Blondeau Da Silva.

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Appendix A: Python Script

Appendix A: Python Script

Using Proposition 3.1, we can determine the terms of \(({P_{N}^{n}}(d))\), for d ∈⟦1,9⟧. To this end, we have created a script with the Python programming language (Python Software Foundation, Python Language Reference, version 3.4. available at http://www.python.org, see Van Rossum (1995)). The implemented function Prob has three parameters: the value of n, the rank N of the wanted term of the sequence and the value ld of the considered leading digit. Here is the used algorithm:

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Silva, S.B.D. An Alternative to the Oversimplifying Benford’s Law in Experimental Fields. Sankhya B 84, 778–808 (2022). https://doi.org/10.1007/s13571-022-00287-0

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