Title: Benford test based on logarithmic property

Authors: Javad Kazemitabar; Jalil Kazemitabar

Addresses: Department of Electrical and Computer Engineering, Babol Noshirvani University of Technology, Shariati Street, Babol, Mazandaran, Iran ' Statistics Department, University of California Los Angeles, Los Angeles, CA 90095, USA

Abstract: The Benford-like behaviour is commonly seen in practical data. That is, the digit frequency more or less has the distribution first introduced by Benford. Despite common belief, however, most datasets do not conform perfectly to Benford's law; they fail famous Benford tests in the literature, or as Nigrini puts it, these tests are too powerful for checking the conformity. We propose a new approach on measuring the deviation of datasets from Benford distribution to determine possible abnormality. We show that the conventional digit frequency tests do not fully absorb the 'significant digit' property. We discuss barriers on the way of auditors in using digit tests mainly when the number of samples is too small or too large. We then propose our method using the logarithmic basis of Benford's law which states the mantissa of the logarithm of all practical numbers should be uniformly distributed. We then test several goodness-of-fit techniques that compare the sample data's mantissa distribution with that of the uniform distribution between zero and one. Our experiment on sample datasets show that Kolmogrov-Smirnov test for uniformity works best for small, medium size and even large records.

Keywords: Benford's law; mantissa; logarithmic test; goodness of fit.

DOI: 10.1504/IJAUDIT.2022.129433

International Journal of Auditing Technology, 2022 Vol.4 No.4, pp.279 - 291

Received: 18 Dec 2021
Accepted: 04 Mar 2022
Published online: 09 Mar 2023 *