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Scale-Distortion Inequalities for Mantissas of Finite Data Sets

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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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

    Arno Berger

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Theodore P. Hill

  3. Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA, 93407, USA

    Kent E. Morrison

Authors
  1. Arno Berger
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  2. Theodore P. Hill
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  3. Kent E. Morrison
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Corresponding author

Correspondence to Kent E. Morrison.

Additional information

The first author was partly supported by a Humboldt research fellowship. The second author was supported in part by the National Security Agency and as a Research Scholar in Residence at California Polytechnic State University.

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Berger, A., Hill, T.P. & Morrison, K.E. Scale-Distortion Inequalities for Mantissas of Finite Data Sets. J Theor Probab 21, 97–117 (2008). https://doi.org/10.1007/s10959-007-0112-z

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  • DOI: https://doi.org/10.1007/s10959-007-0112-z

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