Electronic Communications in Probability

Explicit Bounds for the Approximation Error in Benford's Law

Lutz Dümbgen and Christoph Leuenberger

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Abstract

Benford's law states that for many random variables $X > 0$ its leading digit $D = D(X)$ satisfies approximately the equation $\mathbb{P}(D = d) = \log_{10}(1 + 1/d)$ for $d = 1,2,\ldots,9$. This phenomenon follows from another, maybe more intuitive fact, applied to $Y := \log_{10}X$: For many real random variables $Y$, the remainder $U := Y - \lfloor Y\rfloor$ is approximately uniformly distributed on $[0,1)$. The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of $Y$ or some derivative of it. These bounds are an interesting and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford's law.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 10, 99-112.

Dates
Accepted: 22 February 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233437

Digital Object Identifier
doi:10.1214/ECP.v13-1358

Mathematical Reviews number (MathSciNet)
MR2386066

Zentralblatt MATH identifier
1189.60044

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60F99: None of the above, but in this section

Keywords
Hermite polynomials Gumbel distribution Kuiper distance normal distribution total variation uniform distribution Weibull distribution

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Dümbgen, Lutz; Leuenberger, Christoph. Explicit Bounds for the Approximation Error in Benford's Law. Electron. Commun. Probab. 13 (2008), paper no. 10, 99--112. doi:10.1214/ECP.v13-1358. https://projecteuclid.org/euclid.ecp/1465233437


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References

  • M. Abramowitz, I.A. Stegun (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
  • F. Benford (1938). The law of anomalous numbers. Proc. Amer. Phil. Soc.78, 551-572.
  • Diaconis, Persi. The distribution of leading digits and uniform distribution ${rm mod}$ $1$. Ann. Probability 5 (1977), no. 1, 72–81.
  • Duncan, R. L. On the density of the $k$-free integers. Fibonacci Quart. 7 1969 140–142.
  • Engel, Hans-Andreas; Leuenberger, Christoph. Benford's law for exponential random variables. Statist. Probab. Lett. 63 (2003), no. 4, 361–365.
  • Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren. Concrete mathematics.A foundation for computer science.Second edition.Addison-Wesley Publishing Company, Reading, MA, 1994. xiv+657 pp. ISBN: 0-201-55802-5
  • Hill, Theodore P. A statistical derivation of the significant-digit law. Statist. Sci. 10 (1995), no. 4, 354–363.
  • T.P. Hill (1998). The First Digit Phenomenon. American Scientist 86, 358-363.
  • Hill, Theodore P.; Schürger, Klaus. Regularity of digits and significant digits of random variables. Stochastic Process. Appl. 115 (2005), no. 10, 1723–1743.
  • Jolissaint, Paul. Loi de Benford, relations de récurrence et suites équidistribuées.(French) [Benford's law, recurrence relations and uniformly distributed sequences] Elem. Math. 60 (2005), no. 1, 10–18.
  • Kontorovich, Alex V.; Miller, Steven J. Benford's law, values of $L$-functions and the $3x+1$ problem. Acta Arith. 120 (2005), no. 3, 269–297.
  • Knuth, Donald E. The art of computer programming. Vol. 2.Seminumerical algorithms. Second edition. Addison-Wesley Series in Computer Science and Information Processing. Addison-Wesley Publishing Co., Reading, Mass., 1981. xiii+688 pp. ISBN: 0-201-03822-6
  • Leemis, Lawrence M.; Schmeiser, Bruce W.; Evans, Diane L. Survival distributions satisfying Benford's law. Amer. Statist. 54 (2000), no. 4, 236–241.
  • S.J. Miller and M.J. Nigrini. Order statistics and shifted almost Benford behavior. Preprint (arXiv:math/0601344v2).
  • S.J. Miller and M.J. Nigrini (2007). Benford's Law applied to hydrology data - results and relevance to other geophysical data. Mathematical Geology 39, 469-490.
  • Newcomb, Simon. Note on the Frequency of Use of the Different Digits in Natural Numbers. Amer. J. Math. 4 (1881), no. 1-4, 39–40.
  • M. Nigrini (1996). A Taxpayer Compliance Application of Benford's Law. J. Amer. Taxation Assoc. 18, 72-91.
  • Pinkham, Roger S. On the distribution of first significant digits. Ann. Math. Statist. 32 1961 1223–1230.
  • Raimi, Ralph A. The first digit problem. Amer. Math. Monthly 83 (1976), no. 7, 521–538.
  • Royden, H. L. Real analysis.The Macmillan Co., New York; Collier-Macmillan Ltd., London 1963 xvi+284 pp.