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
base
being
. 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
dependent
random variables. We develop a method for approximating the Benford behavior of a
product of
dependent random variables modeled by a copula
distribution
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.