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Hobza, T and Vajda, I (2001)

On the Newcomb-Benford law in models of statistical data

Revista Matematica Complutense XIV(2), pp. 407-420.

ISSN/ISBN: 1139-1138 DOI: Not available at this time.

Abstract: We consider positive real valued random data X with the decadic representation X = ∑i=-∞ Di 10i and the first significant digit D = D(X) ∈ {1, 2, . . . , 9} of X defined by the condition D = Di≥1, Di+1 = Di+2 = . . . = 0. The data X are said to satisfy the Newcomb-Benford law if P{D = d} = log10 (d+1)/d for all d ∈ {1, 2, . . . , 9}. This law holds for example for the data with log10X uniformly distributed on an interval (m, n) where m and n are integers. We show that if log10 X has a distribution function G(x/σ) on the real line where σ>0 and G(x) has an absolutely continuous density g(x) which is monotone on the intervals (−∞, 0) and (0,∞) then | P{D = d} − log10 (d + 1)/ d| ≤ 2 g(0)/σ. The constant 2 can be replaced by 1 if g(x) = 0 on one of the intervals (−∞, 0), (0,∞). Further, the constant 2g(0) is to be replaced by ∫|g'(x)|dx if instead of the monotonicity we assume absolute integrability of the derivative g'(x).

@article{, title={On the Newcomb-Benford law in models of statistical data.}, author={Hobza, Tom{\'a}s and Vajda, Igor}, journal={Revista Matem{\'a}tica Complutense}, volume={14}, number={2}, pages={407--420}, year={2001}, ISSN={1139-1138}, }

Reference Type: Journal Article

Subject Area(s): Probability Theory, Statistics