Best finite approximations of Benford's Law

TL;DR

This paper characterizes the optimal finite approximations of probability measures, including Benford's Law, under various metrics, providing benchmarks for empirical analysis and extending existing theoretical results.

Contribution

It offers a comprehensive framework for best finitely supported approximations of measures with constraints, specifically applied to Benford's Law and other distributions.

Findings

Identifies best finite approximations for Benford's Law.

Extends known results on measure approximation.

Provides benchmarks for empirical Benford's Law analysis.

Abstract

For arbitrary Borel probability measures with compact support on the real line, characterizations are established of the best finitely supported approximations, relative to three familiar probability metrics (Levy, Kantorovich, and Kolmogorov), given any number of atoms, and allowing for additional constraints regarding weights or positions of atoms. As an application, best (constrained or unconstrained) approximations are identified for Benford's Law (logarithmic distribution of significands) and other familiar distributions. The results complement and extend known facts in the literature; they also provide new rigorous benchmarks against which to evaluate empirical observations regarding Benford's Law.