On the newcomb-benford law in models of statistical data

Abstract

We consider positive real valued random data X with the decadic representation X = P1i=−1 Di 10i and the first significant digit D = D(X) 2 {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 2 {1, 2, . . . , 9}. This law holds for example for the data with log10 X uniformly distributed on an interval (m, n) where m and n are integers. We show that if log10 X has a distribution function G(x/_) on the real line where _ > 0 and G(x) has an absolutely continuous density g(x) which is monotone on the intervals (−1, 0) and (0,1) then ____ P{D = d} − log10 d + 1 d ____ _ 2 g(0) _ . The constant 2 can be replaced by 1 if g(x) = 0 on one of the intervals (−1, 0), (0,1). Further, the constant 2g(0) is to be replaced by R |g0(x)|dx if instead of the monotonicity we assume absolute integrability of the derivative g0(x).