Study on the effect of sample size on type I error, in the first, second and first-two digits excessmad tests

Abstract

The first-two digits ExcessMAD test was created in 2016, allowing to evaluate whether a certain data set conforms to Benford’s Law (BL). The purpose of this study is to explore some questions that remained open: develop the exact and approximate mathematical formulation of the first and second digit ExcessMAD test and study the type I error of these tests when applied to different sample sizes conforming to BL and to the uniform distribution, due to its wide use in accounting data. The importance of this study is to make available to accountants, auditors and researchers the first and second digit ExcessMAD tests, which will make it possible to conduct further investigations involving BL, especially for smaller samples. In addition, the relevance of the type I error analysis stems from the reduction of unnecessary additional studies for the investigation of non-conformity, in the case of the erroneous rejection of the null hypothesis stated as conforming to BL. The application of the second digit ExcessMAD test in the uniform distribution reveals that the close proximity between the uniform and BL distributions can lead to misinterpretations. Based on the exact and approximate mathematical formulations of the three ExcessMAD tests and the use of the Monte Carlo simulation technique, samples were generated in accordance with the BL and uniform distributions, with sizes between 100 and 3,500 elements, which allowed the study of type I error and the comparison of the tests applied to those distributions. This paper seeks to cover three gaps in the literature on ExcessMAD tests. In the previous studies, the following approaches were not found: the exact and approximate mathematical formulation of the first and second digit ExcessMAD tests; the analysis of type I error in these tests and the comparison of such results in the BL and Uniform distributions.