Contributions to the Testing of Benford's Law

Abstract

Benford’s Law is a statistical phenomenon stating that the distribution of leading digits in a set of naturally occurring numbers follows a logarithmic trend, where the distribution of the first digit is P(D1 = d1) = log(1+1/d1), d1 ϵ {1,2, ...,9}. While most commonly used for fraud detection in a variety of areas, including accounting, taxation, and elections, recent work has examined applications within multiple choice testing. Building upon this, we look at test bank data from mathematics and statistics textbooks, and apply three commonly used conformity tests: Pearson’s chi-square, MAD, and SSD, and two simultaneous confidence intervals. From there, we run simulation studies to determine the coverage of each, and propose a new conformity test using linear regression with the inverse of the Benford probability function. Our analysis reveals that the inverse regressionmodel is an improvement upon the chi-square goodness of fit test and the regression model that was previously proposed in 2006 by A.D. Saville; however, still presents some asymptotic issues at large sample sizes. The proposed method is compared to the previously utilized tests through numerical examples.