### Kössler, W, Lenz, H-J and Wang, XD (2023)

#### Some new invariant sum tests and MAD tests for the assessment of Benford's Law

Preprint on ResearchSquare.

**ISSN/ISBN:** Not available at this time.
**DOI:** 10.21203/rs.3.rs-3336839/v1

**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 a universe is not uniformly, but logarithmically distributed. Especially, the first non-zero digit D1 is One with probability P (D1 = 1) = log10 2 ≈ 0.3. There are several tests available for testing Benford, the best known are Pearson’s χ2-test, the Kolmogorov-Smirnov test and the MAD-test suggested by Nigrini (2012). The latter test was enhanced to
significance tests in K¨ossler, Lenz and Wang (2021) and in Cerqueti and Lupi (2021).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 investi-
gated, 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 from the sample size. 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.

**Bibtex:**

```
@misc{,
author = {Wolfgang Kössler and Hans-J. Lenz and Xing David Wang},
title = {Some new invariant sum tests and {MAD} tests for the assessment of Benford's Law},
year = {2023},
doi = {10.21203/rs.3.rs-3336839/v1},
}
```

**Reference Type:** Preprint

**Subject Area(s):** Statistics