A test of significance for Benford’s law based on the Chebyshev distance

Abstract

We show, by means of a numerical simulation, that the asymptotic (n ≥ 100) cumulative distribution function of the Chebyshev distance statistic is well approximated by a log-normal function with parameters μ = −0.6183 and σ = 0.3561 in the null hypothesis that Benford’s law holds. The deviations of the cumulative function observed in Monte Carlo simulations from the empirical one are below 0.5%. This makes the statistical test based on the Chebyshev statistic accurate at a level of 1% when testing Benford’s law for moderately large and large numbers of data points. Test values of the Chebyshev distance as a function of the sample size are also estimated empirically by performing a Monte Carlo simulation in the case of low n (10 ≤ n ≤ 99). The efficacy and power of the goodness-of-fit test based on the Chebyshev estimator are analyzed and compared with those based on the Pearson χ2 and Kolmogorov-Smirnov statistics. Finally, an application of the Chebyshev test to the annual deaths counts by country is discussed.

Journal of Statistical Research 2024, Vol. 58, No. 2, pp. 259-277