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=-∞}^{∞} D_{i} 10^{i} and the first significant digit D = D(X) ∈ {1, 2, . . . , 9} of X defined by the condition D = D_{i}≥1, D_{i+1} = D_{i+2} = . . . = 0. The data X are said to satisfy the Newcomb-Benford law if P{D = d} = log_{10} (d+1)/d for all d ∈ {1, 2, . . . , 9}. This law holds for example for the data with log_{10}X uniformly distributed on an interval (m, n) where m and n are integers. We show that if log_{10} 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} − log_{10} (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).
Bibtex:
@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