An Empirical Non-Parametric Likelihood Family of Data-Based Benford-Like Distributions

A mathematical expression known as Benford's law provides an example of an unexpected relationship among randomly selected first significant digits (FSD). Newcomb (1881), and later Benford (1938), conjectured that FSDs would exhibit a weakly monotonic distribution and proposed a frequency proportional to the logarithmic rule. Unfortunately, the Benford FSD function does not hold for a wide range of scale-invariant multiplicative data. To confront this problem we use information-theoretic methods to develop a data-based family of Benford-like exponential distributions that provide null hypotheses for testing purposes. Two data sets are used to illustrate the performance of generalized Benford-like distributions