### Grendar, M, Judge, G and Schechter, L (2007)

#### An empirical non-parametric likelihood family of data-based Benford-like distributions

Physica A: Statistical Mechanics and its Applications 380, pp. 429-438.

**ISSN/ISBN:** 0378-4371
**DOI:** 10.1016/j.physa.2007.02.062

**Abstract:** A mathematical expression known as Benford’s law provides an example of an unexpected relationship among
randomly selected sequences of first significant digits (FSDs). Newcomb [Note on the frequency of use of the different
digits in natural numbers, Am. J. Math. 4 (1881) 39–40], and later Benford [The law of anomalous numbers, Proc. Am.
Philos. Soc. 78(4) (1938) 551–572], conjectured that FSDs would exhibit a weakly monotonic decreasing distribution and
proposed a frequency proportional to the logarithmic rule. Unfortunately, the Benford FSD function does not hold for a
wide range of scale-invariant multiplicative data. To confront this problem we use information-theoretic methods to
develop a data-based family of alternative Benford-like exponential distributions that provide null hypotheses for testing
purposes. Two data sets are used to illustrate the performance of generalized Benford-like distributions.

**Bibtex:**

```
@article{,
title = "An empirical non-parametric likelihood family of data-based Benford-like distributions",
journal = "Physica A: Statistical Mechanics and its Applications",
volume = "380",
pages = "429--438",
year = "2007",
issn = "0378-4371",
doi = "10.1016/j.physa.2007.02.062",
url = "http://www.sciencedirect.com/science/article/pii/S0378437107001963",
author = "Marian Grendar and George Judge and Laura Schechter",
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Statistics