Resumen de On the Newcomb-Benford law in models of statiscal data
We consider positive real valued random data X with the decadic representation X=Si=-¥¥Di 10i and the first significant digit D=D(X)Î{1,2,...,9} of X defined by the condition D=Di³1, 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}dÎ{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/s) on the real line where s>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)/s. 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 ò|g'(x)|dx if instead of the monotonicity we assume absolute integrability of the derivative g'(x).
