Statistical analysis of fitting Pareto and Weibull distributions with Benford's Law: theoretical approach and empirical evidence

This paper studies the fundamental properties of Benford's Law which investigates the distribution of the first digits' appearance within datasets. The purpose and the usefulness of the research developed within the paper are to identify additional distributions, beyond those already investigated, that conform to the Benford distribution. As a main contribution, we state and prove {with the new approach} that the Pareto distribution and {appropriate constant times Weibull density function}, under some parameter constraint, obey Benford's Law. Further, with the statistical tests and simulation method, we quantify how the fit varies as the parameters of the Pareto distribution change. As Benford's Law is one of the main used approaches for detecting data manipulations and frauds in practice, we use that methodology to consider eventual manipulations in a set of data from the financial reports of three private hospitals operating in Serbia. Moreover, we present the conformity of the Weibull distribution to Benford's Law through the analysis of real-world data, where in the Weibull distribution demonstrates a good fit, {even proof of that conformity is a known result in the literature}. By demonstrating the adherence of Benford's characteristics to the Pareto and Weibull distributions, commonly employed for modeling in various fields, those findings can be utilized in many practical studies.