IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v303y2002i3p387-396.html
   My bibliography  Save this article

On the nature of Benford's Law

Author

Listed:
  • Gottwald, Georg A.
  • Nicol, Matthew

Abstract

We study multiplicative and affine sequences of real numbers defined byN(j+1)=ζ(j)N(j)+η(j),where {ζ(j)} and {η(j)} are sequences of positive real numbers (in the multiplicative case η(j)=0 for all j). We investigate the conditions under which the leading digits k of {N(j)} have the following probability distribution, known as Benford's Law, P(k)=log10((k+1)/k). We present two main results. First, we show that contrary to the usual assumption in the literature, {ζ(j)} does not necessarily need to come from a chaotic or independent random process for Benford's Law to hold. The multiplicative driving force may be a deterministic quasiperiodic or even periodic forcing. Second, we give conditions under which the distribution of the first digits of an affine process displays Benford's Law. Our proofs use techniques from ergodic theory.

Suggested Citation

  • Gottwald, Georg A. & Nicol, Matthew, 2002. "On the nature of Benford's Law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 303(3), pages 387-396.
    • Handle: RePEc:eee:phsmap:v:303:y:2002:i:3:p:387-396
    DOI: 10.1016/S0378-4371(01)00497-6
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437101004976
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/S0378-4371(01)00497-6?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. César Carrera, 2015. "Tracking exchange rate management in Latin America," Review of Financial Economics, John Wiley & Sons, vol. 25(1), pages 35-41, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Shao, Zhi-Gang & Jian-Ping Sang, & Zou, Xian-Wu & Tan, Zhi-Jie & Jin, Zhun-Zhi, 2005. "Blackmail propagation on small-world networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 351(2), pages 662-670.
    2. Villas-Boas, Sofia B. & Fu, Qiuzi & Judge, George, 2017. "Benford’s law and the FSD distribution of economic behavioral micro data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 711-719.
    3. 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).
    4. Hürlimann, Werner, 2015. "On the uniform random upper bound family of first significant digit distributions," Journal of Informetrics, Elsevier, vol. 9(2), pages 349-358.
    5. Matthew A. Cole & David J. Maddison & Liyun Zhang, 2020. "Testing the emission reduction claims of CDM projects using the Benford’s Law," Climatic Change, Springer, vol. 160(3), pages 407-426, June.
    6. Carlos Velarde & Alberto Robledo, 2017. "Rank distributions: Frequency vs. magnitude," PLOS ONE, Public Library of Science, vol. 12(10), pages 1-13, October.
    7. Azevedo, Caio da Silva & Gonçalves, Rodrigo Franco & Gava, Vagner Luiz & Spinola, Mauro de Mesquita, 2021. "A Benford’s Law based methodology for fraud detection in social welfare programs: Bolsa Familia analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    8. Tariq Ahmad Mir, 2012. "The leading digit distribution of the worldwide Illicit Financial Flows," Papers 1201.3432, arXiv.org, revised Nov 2012.
    9. Clippe, Paulette & Ausloos, Marcel, 2012. "Benford’s law and Theil transform of financial data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6556-6567.
    10. Bormashenko, Ed. & Shulzinger, E. & Whyman, G. & Bormashenko, Ye., 2016. "Benford’s law, its applicability and breakdown in the IR spectra of polymers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 524-529.
    11. Whyman, G. & Ohtori, N. & Shulzinger, E. & Bormashenko, Ed., 2016. "Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 595-601.
    12. 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.
    13. Biau, Damien, 2015. "The first-digit frequencies in data of turbulent flows," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 440(C), pages 147-154.
    14. T. Mir, 2016. "The leading digit distribution of the worldwide illicit financial flows," Quality & Quantity: International Journal of Methodology, Springer, vol. 50(1), pages 271-281, January.
    15. Adriano Silva & Sergio Floquet & Ricardo Lima, 2023. "Newcomb–Benford’s Law in Neuromuscular Transmission: Validation in Hyperkalemic Conditions," Stats, MDPI, vol. 6(4), pages 1-19, October.
    16. Horton, Joanne & Krishna Kumar, Dhanya & Wood, Anthony, 2020. "Detecting academic fraud using Benford law: The case of Professor James Hunton," Research Policy, Elsevier, vol. 49(8).
    17. David Giles, 2007. "Benford's law and naturally occurring prices in certain ebaY auctions," Applied Economics Letters, Taylor & Francis Journals, vol. 14(3), pages 157-161.
    18. Ferrer i Cancho, Ramon, 2005. "Decoding least effort and scaling in signal frequency distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(1), pages 275-284.
    19. Malgorzata J. Krawczyk & Mateusz Libirt & Krzysztof Malarz, 2024. "Analysis of scientific cooperation at the international and intercontinental level," Scientometrics, Springer;Akadémiai Kiadó, vol. 129(8), pages 4983-5002, August.
    20. Tariq Ahmad Mir & Marcel Ausloos & Roy Cerqueti, 2014. "Benford's law predicted digit distribution of aggregated income taxes: the surprising conformity of Italian cities and regions," Papers 1410.2890, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:phsmap:v:303:y:2002:i:3:p:387-396. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.