Scale-Distortion Inequalities for Mantissas of Finite Data Sets |
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Authors: | Arno Berger Theodore P Hill Kent E Morrison |
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Institution: | (1) Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand;(2) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;(3) Department of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, USA |
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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.
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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Keywords: | Benford’ s Law Scale-invariance Scale-distortion Mantissa distribution Kantorovich metric |
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