## Electronic Communications in Probability

### Explicit Bounds for the Approximation Error in Benford's Law

#### 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

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

Rights

#### 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

#### 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.