Base Dependence of Benford Random Variables

TL;DR

This paper investigates how the Benford property of a random variable depends on the base, introducing the 'Benford spectrum' and analytical tools to study base dependence, with applications to various families of Benford variables.

Contribution

It introduces the 'Benford spectrum' concept and analytical methods to analyze base dependence of Benford random variables, extending previous work.

Findings

Benford property is generally base-dependent.

Methods to analyze and characterize base dependence are developed.

Examples demonstrate the application of these methods to different Benford distributions.

Abstract

A random variable X that is base b Benford will not in general be base c Benford when c is not equal to b. This paper builds on two of my earlier papers and is an attempt to cast some light on the issue of base dependence. Following some introductory material, the "Benford spectrum" of a positive random variable is introduced and known analytic results about Benford spectra are summarized. Some standard machinery for a "Benford analysis" is introduced and combined with my method of "seed functions" to yield tools to analyze the base c Benford properties of a base b Benford random variable. Examples are generated by applying these general methods to several families of Benford random variables. Berger and Hill's concept of "base-invariant significant digits" is discussed. Some potential extensions are sketched.