Ubiquity of Benford's law and emergence of the reciprocal distribution

Friar, James Lewis; Goldman, Terrance J.; Pérez-Mercader, J.

doi:10.1016/j.physleta.2016.03.045

Title: Ubiquity of Benford's law and emergence of the reciprocal distribution

Abstract

In this paper, we apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. Finally, we separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.

Authors:

Friar, James Lewis ^[1]; Goldman, Terrance J. ^[1];

^[2]

Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division
Harvard Univ., Cambridge, MA (United States). Dept. of Earth and Planetary Sciences; Santa Fe Inst. (SFI), Santa Fe, NM (United States)

Publication Date:: Thu Apr 07 00:00:00 EDT 2016

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE National Nuclear Security Administration (NNSA)

Contributing Org.:: Harvard Univ., Cambridge, MA (United States)

OSTI Identifier:: 1321724

Alternate Identifier(s):: OSTI ID: 1345436

Report Number(s):: LA-UR-13-20486
Journal ID: ISSN 0375-9601

Grant/Contract Number:: AC52-06NA25396

Resource Type:: Accepted Manuscript

Journal Name:: Physics Letters. A

Additional Journal Information:: Journal Volume: 380; Journal Issue: 22-23; Journal ID: ISSN 0375-9601

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Benford/Reciprocal distribution; Information/Communication theory; Iteration theory

Citation Formats


                    Friar, James Lewis, Goldman, Terrance J., and Pérez-Mercader, J. Ubiquity of Benford's law and emergence of the reciprocal distribution.  United States: N. p., 2016. 
Web.  doi:10.1016/j.physleta.2016.03.045.

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                    Friar, James Lewis, Goldman, Terrance J., & Pérez-Mercader, J. Ubiquity of Benford's law and emergence of the reciprocal distribution.  United States.  https://doi.org/10.1016/j.physleta.2016.03.045

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                    Friar, James Lewis, Goldman, Terrance J., and Pérez-Mercader, J. Thu .  
"Ubiquity of Benford's law and emergence of the reciprocal distribution".  United States.  https://doi.org/10.1016/j.physleta.2016.03.045.  https://www.osti.gov/servlets/purl/1321724.

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@article{osti_1321724,

  title        = {Ubiquity of Benford's law and emergence of the reciprocal distribution},

  author       = {Friar, James Lewis and Goldman, Terrance J. and Pérez-Mercader, J.},

  abstractNote = {In this paper, we apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. Finally, we separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.},

  doi          = {10.1016/j.physleta.2016.03.045},

  journal      = {Physics Letters. A},

  number       = 22-23,

  volume       = 380,

  place        = {United States},

  year         = {Thu Apr 07 00:00:00 EDT 2016},

  month        = {Thu Apr 07 00:00:00 EDT 2016}

}

Copy to clipboard

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Works referenced in this record:

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection
book, January 2012

Nigrini, Mark J.
John Wiley & Sons
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Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem
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Benford’s law and cross-sections of A(n, $ \alpha$ )B reactions
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Liu, X. J.; Zhang, X. P.; Ni, D. D.
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Benford’s law and complex atomic spectra
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Pain, Jean-Christophe
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Newcomb-Benford law in Astrophysical Sources
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Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf
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Benford’s law, its applicability and breakdown in the IR spectra of polymers
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Genome Sizes and the Benford Distribution
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Friar, James L.; Goldman, Terrance; Pérez–Mercader, Juan
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Journal of Computational Physics, Vol. 28, Issue 3
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Similar Records

Title: Ubiquity of Benford's law and emergence of the reciprocal distribution

Abstract

Citation Formats

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection book, January 2012

Note on the Frequency of Use of the Different Digits in Natural Numbers journal, January 1881

A Statistical Derivation of the Significant-Digit Law journal, November 1995

Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem journal, February 2011

A basic theory of Benford’s Law journal, January 2011

Benford’s law and cross-sections of A(n, $ \alpha$ )B reactions journal, June 2011

Benford’s law and complex atomic spectra journal, January 2008

Newcomb-Benford law in Astrophysical Sources journal, November 2006

Benford's law in the natural sciences journal, November 2010

Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf journal, April 2001

Benford’s law, its applicability and breakdown in the IR spectra of polymers journal, February 2016

Genome Sizes and the Benford Distribution journal, May 2012

On the Distribution of Numbers journal, October 1970

On the Distribution of First Significant Digits journal, December 1961

A Mathematical Theory of Communication journal, July 1948

A note on the roundoff error in the Numerov algorithm journal, September 1978

Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection
book, January 2012

Note on the Frequency of Use of the Different Digits in Natural Numbers
journal, January 1881

A Statistical Derivation of the Significant-Digit Law
journal, November 1995

Benford’s Law Strikes Back: No Simple Explanation in Sight for Mathematical Gem
journal, February 2011

A basic theory of Benford’s Law
journal, January 2011

Benford’s law and cross-sections of A(n, $ \alpha$ )B reactions
journal, June 2011

Benford’s law and complex atomic spectra
journal, January 2008

Newcomb-Benford law in Astrophysical Sources
journal, November 2006

Benford's law in the natural sciences
journal, November 2010

Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf
journal, April 2001

Benford’s law, its applicability and breakdown in the IR spectra of polymers
journal, February 2016

Genome Sizes and the Benford Distribution
journal, May 2012

On the Distribution of Numbers
journal, October 1970

On the Distribution of First Significant Digits
journal, December 1961

A Mathematical Theory of Communication
journal, July 1948

A note on the roundoff error in the Numerov algorithm
journal, September 1978