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

  • Grendar, Marian
  • Judge, George
  • Schechter, Laura

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.

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File URL: http://www.sciencedirect.com/science/article/pii/S0378437107001963
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Article provided by Elsevier in its journal Physica A: Statistical Mechanics and its Applications.

Volume (Year): 380 (2007)
Issue (Month): C ()
Pages: 429-438

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Handle: RePEc:eee:phsmap:v:380:y:2007:i:c:p:429-438
DOI: 10.1016/j.physa.2007.02.062
Contact details of provider: Web page: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/

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  1. Hill, Theodore P. & Schürger, Klaus, 2005. "Regularity of digits and significant digits of random variables," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1723-1743, October.
  2. repec:cup:cbooks:9780521623940 is not listed on IDEAS
  3. Scott Marchi & James Hamilton, 2006. "Assessing the Accuracy of Self-Reported Data: an Evaluation of the Toxics Release Inventory," Journal of Risk and Uncertainty, Springer, vol. 32(1), pages 57-76, January.
  4. Pietronero, L. & Tosatti, E. & Tosatti, V. & Vespignani, A., 2001. "Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 293(1), pages 297-304.
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