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N. Mumic, P. Filzmoser:
"A multivariate test for detecting fraud based on Benford´s law, with application to music streaming data";
Statistical Methods & Applications, 30 (2021), S. 819 - 840.

Benford´s law became a prevalent concept for fraud and anomaly detection. It examines the frequencies of the leading digits of numbers in a collection of data and states that the leading digit is most often 1, with diminishing frequencies up to 9. In this paper we propose a multivariate approach to test whether the observed frequencies follow the theoretical Benford distribution. Our approach is based on the
concept of compositional data, which examines the relative information between the frequencies of the leading digits. As a result, we introduce a multivariate test for Benford distribution. In simulation studies and examples we compare the multivariate test performance to the conventional chi-square and Kolmogorov-Smirnov test, where the multivariate test turns out to be more sensitive in many cases. A
diagnostics plot based on relative information allows to reveal and interpret the possible deviations from the Benford distribution.

http://dx.doi.org/10.1007/s10260-021-00582-6

https://link.springer.com/content/pdf/10.1007/s10260-021-00582-6.pdf