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Berger, A, Hill, TP and Morrison, KE (2008)

Scale-Distortion Inequalities for Mantissas of Finite Data Sets

Journal of Theoretical Probability 21(1), pp. 97-117.

ISSN/ISBN: 0894-9840 DOI: Not available at this time.

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 scaledistortion 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.

@article {, AUTHOR = {Berger, Arno and Hill, Theodore P. and Morrison, Kent}, TITLE = {Scale-Distortion Inequalities for Mantissas of Finite Data Sets}, JOURNAL = {Journal of Theoretical Probability}, VOLUME = {21}, NUMBER = {1}, YEAR = {2008}, PAGES = {97--117}, ISSN = {0894-9840}, }

Reference Type: Journal Article

Subject Area(s): Probability Theory