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Scale-Distortion Inequalities for Mantissas of Finite Data Sets

Author

Listed:
  • Arno Berger

    (University of Canterbury)

  • Theodore P. Hill

    (Georgia Institute of Technology)

  • Kent E. Morrison

    (California Polytechnic State University)

Abstract

In scientific computations using floating point arithmetic, rescaling a data set multiplicatively (e.g., corresponding to a conversion from dollars to euros) changes the distribution of the mantissas, or fraction parts, of the data. A scale-distortion factor for probability distributions is defined, based on the Kantorovich distance between distributions. Sharp lower bounds are found for the scale-distortion of n-point data sets, and the unique data set of size n with the least scale-distortion is identified for each positive integer n. A sequence of real numbers is shown to follow Benford’s Law (base b) if and only if the scale-distortion (base b) of the first n data points tends zero as n goes to infinity. These results complement the known fact that Benford’s Law is the unique scale-invariant probability distribution on mantissas.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jotpro:v:21:y:2008:i:1:d:10.1007_s10959-007-0112-z
    DOI: 10.1007/s10959-007-0112-z
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Arno Berger & Chuang Xu, 2020. "Asymptotics of One-Dimensional Lévy Approximations," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1164-1195, June.
    2. Arno Berger & Chuang Xu, 2019. "Best Finite Approximations of Benford’s Law," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1525-1553, September.

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