ON THE NEWCOMB-BENFORD LAW IN MODELS OF STATISTICAL DATA ∗

We consider positive real valued random data X with the decadic representation X = i=−∞Di 10 i and the first signifi-cant digit D = D(X) ∈ {1, 2,..., 9} of X defined by the con-dition 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 ∈ {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 dis-tribution function G(x/σ) on the real line where σ> 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+ 1d ∣∣∣ ∣ ≤ 2 g(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).