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Explicit Bounds for the Approximation Error in Benford's Law


 
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1. Title Title of document Explicit Bounds for the Approximation Error in Benford's Law
 
2. Creator Author's name, affiliation, country Lutz Dümbgen; University of Bern
 
2. Creator Author's name, affiliation, country Christoph Leuenberger; Ecole d'Ingénieurs de Fribourg
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Hermite polynomials, Gumbel distribution, Kuiper distance, normal distribution, total variation, uniform distribution, Weibull distribution
 
3. Subject Subject classification 60E15, 60F99
 
4. Description 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.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Swiss National Science Foundation
 
7. Date (YYYY-MM-DD) 2008-02-22
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1358
 
10. Identifier Digital Object Identifier 10.1214/ECP.v13-1358
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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