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Benford's Law, Families of Distributions and a Test Basis

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  • John Morrow

Abstract

Benford's Law is used to test for data irregularities. While novel, there are two weaknesses in the current methodology. First, test values used in practice are too conservative and the test values of this paper are more powerful and hold for fairly small samples. Second, testing requires Benford's Law to hold, which it often does not. I present a simple method to transform distributions to satisfy the Law with arbitrary precision and induce scale invariance, freeing tests from the choice of units. I additionally derive a rate of convergence to Benford's Law. Finally, the results are applied to common distributions.

Suggested Citation

  • John Morrow, 2014. "Benford's Law, Families of Distributions and a Test Basis," CEP Discussion Papers dp1291, Centre for Economic Performance, LSE.
    • Handle: RePEc:cep:cepdps:dp1291
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    References listed on IDEAS

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    1. Tam Cho, Wendy K. & Gaines, Brian J., 2007. "Breaking the (Benford) Law: Statistical Fraud Detection in Campaign Finance," The American Statistician, American Statistical Association, vol. 61, pages 218-223, August.
    2. G. Noether, 1963. "Note on the kolmogorov statistic in the discrete case," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 7(1), pages 115-116, December.
    3. Grendar, Marian & Judge, George & Schechter, Laura, 2007. "An empirical non-parametric likelihood family of data-based Benford-like distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 429-438.
    4. Scott Marchi & James Hamilton, 2006. "Assessing the Accuracy of Self-Reported Data: an Evaluation of the Toxics Release Inventory," Journal of Risk and Uncertainty, Springer, vol. 32(1), pages 57-76, January.
    5. Rodriguez R.J., 2004. "First Significant Digit Patterns From Mixtures of Uniform Distributions," The American Statistician, American Statistical Association, vol. 58, pages 64-71, February.
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    Cited by:

    1. Rabeea Sadaf, 2017. "Advanced Statistical Techniques For Testing Benford'S Law," Annals of Faculty of Economics, University of Oradea, Faculty of Economics, vol. 1(2), pages 229-238, December.
    2. Bernhard Rauch & Max Göttsche & Gernot Brähler & Stefan Engel, 2011. "Fact and Fiction in EU‐Governmental Economic Data," German Economic Review, Verein für Socialpolitik, vol. 12(3), pages 243-255, August.
    3. Ronelle Burger & Canh Thien Dang & Trudy Owens, 2017. "Better performing NGOs do report more accurately: Evidence from investigating Ugandan NGO financial accounts," Discussion Papers 2017-10, University of Nottingham, CREDIT.
    4. Dang, Canh Thien & Owens, Trudy, 2020. "Does transparency come at the cost of charitable services? Evidence from investigating British charities," Journal of Economic Behavior & Organization, Elsevier, vol. 172(C), pages 314-343.
    5. Holz, Carsten A., 2014. "The quality of China's GDP statistics," China Economic Review, Elsevier, vol. 30(C), pages 309-338.
    6. Kalaichelvan, Mohandass & Lim Kai Jie, Shawn, 2012. "A Critical Evaluation of the Significance of Round Numbers in European Equity Markets in Light of the Predictions from Benford’s Law," MPRA Paper 40960, University Library of Munich, Germany.
    7. Eutsler, Jared & Kathleen Harris, M. & Tyler Williams, L. & Cornejo, Omar E., 2023. "Accounting for partisanship and politicization: Employing Benford's Law to examine misreporting of COVID-19 infection cases and deaths in the United States," Accounting, Organizations and Society, Elsevier, vol. 108(C).
    8. George Judge & Laura Schechter, 2009. "Detecting Problems in Survey Data Using Benford’s Law," Journal of Human Resources, University of Wisconsin Press, vol. 44(1).
    9. Hürlimann, Werner, 2015. "On the uniform random upper bound family of first significant digit distributions," Journal of Informetrics, Elsevier, vol. 9(2), pages 349-358.
    10. Pankaj C. Patel & Mike G. Tsionas & Maria João Guedes, 2022. "Benford's law, small business financial reporting, and survival," Managerial and Decision Economics, John Wiley & Sons, Ltd., vol. 43(8), pages 3301-3315, December.
    11. Druică, Elena & Oancea, Bogdan & Vâlsan, Călin, 2018. "Benford's law and the limits of digit analysis," International Journal of Accounting Information Systems, Elsevier, vol. 31(C), pages 75-82.
    12. Thomas Stoerk, 2015. "Statistical corruption in Beijing’s air quality data has likely ended in 2012," GRI Working Papers 194, Grantham Research Institute on Climate Change and the Environment.

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    More about this item

    Keywords

    Benfords Law; data quality; fraud detection;
    All these keywords.

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C24 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Truncated and Censored Models; Switching Regression Models; Threshold Regression Models
    • C46 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Specific Distributions

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