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DŁmbgen, L and Leuenberger, C (2008)

Explicit Bounds for the Approximation Error in Benfordís Law

Electronic Communications in Probability 13, pp. 99-112.

ISSN/ISBN: 1083-589X DOI: 10.1214/ECP.v13-1358



Abstract: Benfordís law states that for many random variables X > 0 its leading digit D = D(X) satisfies approximately the equation P(D = d) = log10 (1 + 1/d) for d = 1, 2, . . . , 9. This phenomenon follows from another, maybe more intuitive fact, applied to Y := log10 X: For many real random variables Y , the remainder U := Y − [Y] 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 Benfordís law.


Bibtex:
@article{, author = "Dümbgen, Lutz and Leuenberger, Christoph", doi = "10.1214/ECP.v13-1358", fjournal = "Electronic Communications in Probability", journal = "Electron. Commun. Probab.", pages = "99--112", pno = "10", publisher = "The Institute of Mathematical Statistics and the Bernoulli Society", title = "Explicit Bounds for the Approximation Error in Benford's Law", url = "https://doi.org/10.1214/ECP.v13-1358", volume = "13", year = "2008" }


Reference Type: Journal Article

Subject Area(s): Probability Theory