We consider positive real valued random data X with the decadic representation X = Σi=∞ ∞Di 10i and the first significant digit D = D(X) in {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 in {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 log10X 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)| ≤ 2g(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{urn:eudml:doc:44389,
title = {On the Newcomb-Benford law in models of statistical data.},
journal = {Revista Matem\'atica de la Universidad Complutense de Madrid},
volume = {14},
year = {2001},
pages = {407-420},
zbl = {1006.62010},
mrnumber = {MR1871305},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:44389}
}
Hobza, Tomás; Vajda, Igor. On the Newcomb-Benford law in models of statistical data.. Revista Matemática de la Universidad Complutense de Madrid, Tome 14 (2001) pp. 407-420. http://gdmltest.u-ga.fr/item/urn:eudml:doc:44389/