Proceedings of International Conference on Management Engineering, Software Engineering and Service Sciences (ICMSS 2022), N. Wahi et al. (Eds.), ACSR 98, pp. 195–204.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

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**Abstract:** It was anticipated more than a century ago that the distribution of first digits in real-world observations would not be uniform, instead follow a trend in which measurements with lower first digits occur more frequently than measurements with higher first digits. Frank Benford coined the term “First Digit Phenomena” to describe this phenomenon, which is now known as Benford’s Law distribution. Benford’s Law distribution has long been recognized but was widely dismissed as a mathematical oddity in the natural sciences. There is a theoretical requirement to analyze such disparities as departures from Benford’s Law have been observed. The use of parametric extensions to existing Benford’s Law is justified, as evidenced by the inclusion of k-tuples as a new parameter in the study. A k-tuples can be interpreted as a set of order and cardinality of first significant leading digit in datasets. Therefore, a convenience and concise method for deriving parametric analytical expansions of Benford’s Law for first significant leading digits is proposed by embedding k-tuples. A new probabilistic explanation for the appearance of extended Benford’s Law distribution has been discovered. As a result, a one-parameter analytical extension of Benford’s Law for first significant leading digits is proposed. The new distribution generated by embedding k-tuples is scale invariant and robust to existing Benford’s Law properties which a sum of first digit proportion is equal to 1, unimodality, logarithmic distribution and positive skewness. Then, mathematical features are investigated and a new generic class of moments generating functions is created. Based on natural phenomenon number, extended Benford’s Law shows lesser values than existing one. This study found that the extended Benford’s Law distribution to be better than the existing Benford’s Law with measurements of lower digit occur more frequently.

**Bibtex:**

```
@inproceedings{,
AUTHOR={Shar Nizam Sharif and Saiful Hafizah Jaaman-Sharman},
TITLE={Robustness of Extended Benford’s Law Distribution and Its Properties},
BOOKTITLE={Proceedings of the 2022 International Conference on Management Engineering, Software Engineering and Service Sciences (ICMSS},
ADDRESS={},
MONTH={},
YEAR={2022},
URL={https://www.atlantis-press.com/article/125978188.pdf},
PAGES = {195--204},
}
```

**Reference Type:** Conference Paper

**Subject Area(s):** Computer Science, Probability Theory