Benford’s law beyond independence : tracking Benford behavior in copula models

Durst, Rebecca F.; Miller, Steven J.

doi:10.2140/involve.2019.12.1193

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Benford's law beyond independence: tracking Benford behavior in copula models

Rebecca F. Durst and Steven J. Miller

Vol. 12 (2019), No. 7, 1193–1218

DOI: 10.2140/involve.2019.12.1193

Abstract

Benford’s law describes a common phenomenon among many naturally occurring data sets and distributions in which the leading digits of the data are distributed with the probability of a first digit of $d$ base $B$ being ${log}_{B} ((d + 1) ∕ d)$ . As it often successfully detects fraud in medical trials, voting, science and finance, significant effort has been made to understand when and how distributions exhibit Benford behavior. Most of the previous work has been restricted to cases of independent variables, and little is known about situations involving dependence. We use copulas to investigate the Benford behavior of the product of $n$ dependent random variables. We develop a method for approximating the Benford behavior of a product of $n$ dependent random variables modeled by a copula distribution $C$ and quantify and bound a copula distribution’s distance from Benford behavior. We then investigate the Benford behavior of various copulas under varying dependence parameters and number of marginals. Our investigations show that the convergence to Benford behavior seen with independent random variables as the number of variables in the product increases is not necessarily preserved when the variables are dependent and modeled by a copula. Furthermore, there is strong indication that the preservation of Benford behavior of the product of dependent random variables may be linked more to the structure of the copula than to the Benford behavior of the marginal distributions.

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Keywords

Benford's law, probability, theoretical statistics

Mathematical Subject Classification 2010

Primary: 11K99, 60E99

Milestones

Received: 16 January 2019

Revised: 11 April 2019

Accepted: 15 April 2019

Published: 12 October 2019

Communicated by Stephan Garcia

Authors

Rebecca F. Durst
Department of Mathematics and Statistics Williams College Williamstown, MA United States	Division of Applied Mathematics Brown University Providence, RI United States

Steven J. Miller
Department of Mathematics and Statistics Williams College Williamstown, MA United States	Department of Mathematics Carnegie Mellon University Pittsburgh, PA United States

Vol. 12, No. 7, 2019