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On the Newcomb-Benford law in models of statistical data. (English) Zbl 1006.62010
Summary: We consider positive, real valued random data $$X$$ with the decadic representation $$X=\sum^\infty_{i= -\infty} D_i10^i$$ and the first significant digit $$D=D(X)\in \{1,2,\dots, 9\}$$ of $$X$$ defined by the condition $$D=D_i\geq 1$$, $$D_{i+1}= D_{i+2}= \cdots=0$$. The data $$X$$ are said to satisfy the Newcomb-Benford law if $$P\{D=d\}= \log_{10}(d+1)/d$$ for all $$d\in\{1,2, \dots, 9 \}$$. 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 $$G(x/\sigma)$$ on the real line, where $$\sigma>0$$ and $$G(x)$$ has an absolutely continuous density $$g(x)$$ which is monotone on the intervals $$(-\infty,0)$$ and $$(0,\infty)$$, then $\bigl|P\{D=d\}-\log_{10} (d+1)/d \bigr|\leq 2g(0)/ \sigma.$ The constant 2 can be replaced by 1 if $$g(x)=0$$ on one of the intervals $$(-\infty)$$, $$(0,\infty)$$. Further, the constant $$2g(0)$$ is to be replaced by $$\int|g'(x)|dx$$ if instead of the monotonicity we assume absolute integrability of the derivative $$g'(x)$$.
##### MSC:
 62E20 Asymptotic distribution theory in statistics 60C05 Combinatorial probability 62-07 Data analysis (statistics) (MSC2010)
Newcomb law
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