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Scale-Distortion Inequalities for Mantissas of Finite Data Sets
Journal of Theoretical Probability
  • Arno Berger, University of Canterbury
  • Theodore P. Hill, Georgia Institute of Technology - Main Campus
  • Kent E. Morrison, California Polytechnic State University - San Luis Obispo
Publication Date

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.

Citation Information
Arno Berger, Theodore P. Hill and Kent E. Morrison. "Scale-Distortion Inequalities for Mantissas of Finite Data Sets" Journal of Theoretical Probability Vol. 21 (2008) p. 97 - 117
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