Electronic Communications in Probability

Explicit Bounds for the Approximation Error in Benford's Law

Lutz Dümbgen and Christoph Leuenberger

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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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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