Some New Tests of Conformity with Benford’s Law

Roy Cerqueti,Claudio Lupi

doi:10.3390/stats4030044

Roy Cerqueti, Claudio Lupi

Open Access

https://doi.org/10.3390/stats4030044

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Journal: Stats	Publication Date: Sep 6, 2021
Citations: 9	License type: CC BY 4.0

Affiliation: Sapienza University of Rome, University of Molise

Abstract

This paper presents new perspectives and methodological instruments for verifying the validity of Benford’s law for a large given dataset. To this aim, we first propose new general tests for checking the statistical conformity of a given dataset with a generic target distribution; we also provide the explicit representation of the asymptotic distributions of the relevant test statistics. Then, we discuss the applicability of such novel devices to the case of Benford’s law. We implement extensive Monte Carlo simulations to investigate the size and the power of the introduced tests. Finally, we discuss the challenging theme of interpreting, in a statistically reliable way, the conformity between two distributions in the presence of a large number of observations.

Highlights

IntroductionWe advance some new tests for verifying the compliance of the empirical distribution obtained from a given population
We present a test based on Wald’s statistic and a new version of a mean absolute deviation (MAD)-based test
The ordering of the tests is the same as in the first digit case; the tests are generally more powerful in the first digit case. These results suggest that in applications it is generally a good idea not to rely on a single test, but to use a battery of different tests designed to detect particular deviations from the null

Summary

Introduction

We advance some new tests for verifying the compliance of the empirical distribution obtained from a given population In this respect, we mention the recent contribution [32], where the authors suggested a statistical test based on the mean. Kossovsky’s criticism ([12], in this Special Issue), where the author refers to the “mistaken use of the Chi-Square test in Benford’s law” From this perspective, we mention the theme of the selection of the critical thresholds for having perfect/marginal/acceptable conformity with Benford’s law (see [3,4] and the recent study developed by [33]).

New Tests of Conformity with Benford’s Law

Monte Carlo Simulations

First-Digit Law

First Two Digits Law

Statistical versus Practical Significance

Findings

Conclusions