# On the Newcomb-Benford law in models of statistical data.

• Volume: 14, Issue: 2, page 407-420
• ISSN: 1139-1138

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## Abstract

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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&gt;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).

## How to cite

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Hobza, Tomás, and Vajda, Igor. "On the Newcomb-Benford law in models of statistical data.." Revista Matemática Complutense 14.2 (2001): 407-420. <http://eudml.org/doc/44389>.

@article{Hobza2001,
abstract = {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&gt;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).},
author = {Hobza, Tomás, Vajda, Igor},
journal = {Revista Matemática Complutense},
keywords = {Distribución asintótica; Distribución logarítmica; Análisis de datos; Newcomb law},
language = {eng},
number = {2},
pages = {407-420},
title = {On the Newcomb-Benford law in models of statistical data.},
url = {http://eudml.org/doc/44389},
volume = {14},
year = {2001},
}

TY - JOUR
AU - Hobza, Tomás
AU - Vajda, Igor
TI - On the Newcomb-Benford law in models of statistical data.
JO - Revista Matemática Complutense
PY - 2001
VL - 14
IS - 2
SP - 407
EP - 420
AB - 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&gt;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).
LA - eng
KW - Distribución asintótica; Distribución logarítmica; Análisis de datos; Newcomb law
UR - http://eudml.org/doc/44389
ER -

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