Vol. 12, No. 7, 2019

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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
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 logB((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.

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