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Some new invariant sum tests and MAD tests for the assessment of Benford’s law

Author

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  • Wolfgang Kössler

    (Humboldt Universität zu Berlin)

  • Hans-J. Lenz

    (Freie Universität Berlin)

  • Xing D. Wang

    (Humboldt Universität zu Berlin)

Abstract

The Benford law is used world-wide for detecting non-conformance or data fraud of numerical data. It says that the significand of a data set from the universe is not uniformly, but logarithmically distributed. Especially, the first non-zero digit is One with an approximate probability of 0.3. There are several tests available for testing Benford, the best known are Pearson’s $$\chi ^2$$ χ 2 -test, the Kolmogorov–Smirnov test and a modified version of the MAD-test. In the present paper we propose some tests, three of the four invariant sum tests are new and they are motivated by the sum invariance property of the Benford law. Two distance measures are investigated, Euclidean and Mahalanobis distance of the standardized sums to the orign. We use the significands corresponding to the first significant digit as well as the second significant digit, respectively. Moreover, we suggest inproved versions of the MAD-test and obtain critical values that are independent of the sample sizes. For illustration the tests are applied to specifically selected data sets where prior knowledge is available about being or not being Benford. Furthermore we discuss the role of truncation of distributions.

Suggested Citation

  • Wolfgang Kössler & Hans-J. Lenz & Xing D. Wang, 2024. "Some new invariant sum tests and MAD tests for the assessment of Benford’s law," Computational Statistics, Springer, vol. 39(7), pages 3779-3800, December.
    • Handle: RePEc:spr:compst:v:39:y:2024:i:7:d:10.1007_s00180-024-01463-8
    DOI: 10.1007/s00180-024-01463-8
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    References listed on IDEAS

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    1. Roy Cerqueti & Claudio Lupi, 2021. "Some New Tests of Conformity with Benford’s Law," Stats, MDPI, vol. 4(3), pages 1-17, September.
    2. Andreas Diekmann, 2007. "Not the First Digit! Using Benford's Law to Detect Fraudulent Scientif ic Data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 34(3), pages 321-329.
    3. Liu, Huan & Tang, Yongqiang & Zhang, Hao Helen, 2009. "A new chi-square approximation to the distribution of non-negative definite quadratic forms in non-central normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 853-856, February.
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