Benford's Law Beyond Independence: Tracking Benford Behavior in Copula Models

TL;DR

This paper explores how dependence among variables, modeled by copulas, affects the emergence of Benford's law in the product of those variables, revealing that dependence can alter expected Benford behavior.

Contribution

It introduces a method to approximate and analyze Benford behavior in dependent variables using copulas, extending understanding beyond independent cases.

Findings

Dependence can disrupt the convergence to Benford's law in products of variables.

The structure of the copula influences Benford behavior more than marginal distributions.

Benford behavior preservation depends on copula structure, not just marginals.

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 $\log_{B}{\frac{d+1}{d}}$. 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.