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2008 Explicit Bounds for the Approximation Error in Benford's Law
Lutz Dümbgen, Christoph Leuenberger
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Electron. Commun. Probab. 13: 99-112 (2008). DOI: 10.1214/ECP.v13-1358


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.


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Lutz Dümbgen. Christoph Leuenberger. "Explicit Bounds for the Approximation Error in Benford's Law." Electron. Commun. Probab. 13 99 - 112, 2008.


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

zbMATH: 1189.60044
MathSciNet: MR2386066
Digital Object Identifier: 10.1214/ECP.v13-1358

Primary: 60E15
Secondary: 60F99

Keywords: Gumbel distribution , Hermite polynomials , Kuiper distance , normal distribution , Total variation , uniform distribution , Weibull distribution

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