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.

前两位数 ExcessMAD 测试创建于 2016 年，用于评估某个数据集是否符合本福德定律 (BL)。本研究的目的是探索一些仍然悬而未决的问题：开发第一位和第二位 ExcessMAD 测试的精确和近似数学公式，并研究这些测试在应用于符合 BL 和均匀分布，因为它在会计数据中的广泛使用。这项研究的重要性在于向会计师、审计师和研究人员提供第一位和第二位 ExcessMAD 测试，这将使得有可能进行涉及 BL 的进一步调查，特别是对于较小的样本。此外，I 类错误分析的相关性源于在错误拒绝声明为符合 BL 的原假设的情况下，减少了为调查不符合项而进行的不必要的额外研究。第二位 ExcessMAD 测试在均匀分布中的应用表明，均匀分布和 BL 分布之间的接近可能导致误解。基于三个 ExcessMAD 测试的精确和近似数学公式以及蒙特卡罗模拟技术的使用，根据 BL 和均匀分布生成样本，样本大小在 100 到 3,500 个元素之间，这使得 I 类研究成为可能错误和应用于这些分布的测试的比较。本文旨在弥补 ExcessMAD 测试文献中的三个空白。在以前的研究中，没有发现以下方法：第一位和第二位 ExcessMAD 测试的精确和近似的数学公式；这些测试中 I 类错误的分析以及 BL 和均匀分布中此类结果的比较。