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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)
Keywords:
Newcomb law
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