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

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

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.

Suggested Citation

  • Grendar, Marian & Judge, George & Schechter, Laura, 2007. "An empirical non-parametric likelihood family of data-based Benford-like distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 429-438.
  • Handle: RePEc:eee:phsmap:v:380:y:2007:i:c:p:429-438
    DOI: 10.1016/j.physa.2007.02.062
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    References listed on IDEAS

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    1. 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.
    2. 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.
    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.
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    Cited by:

    1. repec:eee:phsmap:v:486:y:2017:i:c:p:711-719 is not listed on IDEAS
    2. Sofia B. Villas-Boas & Qiuzi Fu & George Judge, 2015. "Is Benford’s Law a Universal Behavioral Theory?," Econometrics, MDPI, Open Access Journal, vol. 3(4), pages 1-11, October.
    3. Lee, Joanne & Cho, Wendy K. Tam & Judge, George G, 2009. "Stigler's approach to recovering the distribution of first significant digits in natural data sets," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt9745m98d, Department of Agricultural & Resource Economics, UC Berkeley.
    4. John Morrow, 2014. "Benford's Law, Families of Distributions and a Test Basis," CEP Discussion Papers dp1291, Centre for Economic Performance, LSE.
    5. Lee, Joanne & Judge, George G., 2008. "Identifying falsified clinical data," CUDARE Working Papers 47001, University of California, Berkeley, Department of Agricultural and Resource Economics.
    6. Lee, Joanne & Cho, Wendy K. Tam & Judge, George G., 2010. "Stigler's approach to recovering the distribution of first significant digits in natural data sets," Statistics & Probability Letters, Elsevier, vol. 80(2), pages 82-88, January.
    7. Paul Hofmarcher & Kurt Hornik, 2013. "First Significant Digits and the Credit Derivative Market During the Financial Crisis," Contemporary Economics, University of Finance and Management in Warsaw, vol. 7(2), June.
    8. Lee, Joanne & Judge, George G, 2008. "Identifying falsified clinical data," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt8x00h1c1, Department of Agricultural & Resource Economics, UC Berkeley.
    9. George Judge & Laura Schechter, 2009. "Detecting Problems in Survey Data Using Benford’s Law," Journal of Human Resources, University of Wisconsin Press, vol. 44(1).
    10. Lee, Joanne & Cho, Wendy K. & Judge, George G., 2009. "Stigler's approach to recovering the distribution of first significant digits in natural data sets," CUDARE Working Papers 47000, University of California, Berkeley, Department of Agricultural and Resource Economics.

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