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Best Finite Approximations of Benford’s Law

Author

Listed:
  • Arno Berger

    (University of Alberta)

  • Chuang Xu

    (University of Alberta)

Abstract

For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Lévy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford’s Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; they also provide new rigorous benchmarks against which to evaluate empirical observations regarding Benford’s law.

Suggested Citation

  • Arno Berger & Chuang Xu, 2019. "Best Finite Approximations of Benford’s Law," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1525-1553, September.
    • Handle: RePEc:spr:jotpro:v:32:y:2019:i:3:d:10.1007_s10959-018-0827-z
    DOI: 10.1007/s10959-018-0827-z
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    References listed on IDEAS

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    1. Georg Ch. Pflug & Alois Pichler, 2011. "Approximations for Probability Distributions and Stochastic Optimization Problems," International Series in Operations Research & Management Science, in: Marida Bertocchi & Giorgio Consigli & Michael A. H. Dempster (ed.), Stochastic Optimization Methods in Finance and Energy, edition 1, chapter 0, pages 343-387, Springer.
    2. Arno Berger & Theodore P. Hill & Kent E. Morrison, 2008. "Scale-Distortion Inequalities for Mantissas of Finite Data Sets," Journal of Theoretical Probability, Springer, vol. 21(1), pages 97-117, March.
    3. Steven J. Miller, 2015. "Benford's Law: Theory and Applications," Economics Books, Princeton University Press, edition 1, number 10527.
    4. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
    5. S. Dereich & C. Vormoor, 2011. "The High Resolution Vector Quantization Problem with Orlicz Norm Distortion," Journal of Theoretical Probability, Springer, vol. 24(2), pages 517-544, June.
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