Scale-Distortion Inequalities for Mantissas of Finite Data Sets

Arno Berger (), Theodore P. Hill () and Kent E. Morrison ()
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Arno Berger: University of Canterbury
Theodore P. Hill: Georgia Institute of Technology
Kent E. Morrison: California Polytechnic State University

Journal of Theoretical Probability, 2008, vol. 21, issue 1, 97-117

Abstract: 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.

Keywords: Benford’s Law; Scale-invariance; Scale-distortion; Mantissa distribution; Kantorovich metric (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s10959-007-0112-z

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