Abstract
Benford's law states that for many random variables $X > 0$ its leading digit $D = D(X)$ satisfies approximately the equation $\mathbb{P}(D = d) = \log_{10}(1 + 1/d)$ for $d = 1,2,\ldots,9$. This phenomenon follows from another, maybe more intuitive fact, applied to $Y := \log_{10}X$: For many real random variables $Y$, the remainder $U := Y - \lfloor Y\rfloor$ is approximately uniformly distributed on $[0,1)$. The present paper provides new explicit bounds for the latter approximation in terms of the total variation of the density of $Y$ or some derivative of it. These bounds are an interesting and powerful alternative to Fourier methods. As a by-product we obtain explicit bounds for the approximation error in Benford's law.
Highlights
The First Digit Law is the empirical observation that in many tables of numerical data the leading significant digits are not uniformly distributed as one might suspect at first
The following law was first postulated by Simon Newcomb (1881): Prob(leading digit = d) = log10(1 + 1/d) for d = 1, . . . , 9
The special case of exponentially distributed random variables was studied by Engel and Leuenberger (2003): Such random variables satisfy the first digit law only approximatively, but precise estimates can be given; see Miller and Nigrini (2006) for an alternative proof and extensions
Summary
The First Digit Law is the empirical observation that in many tables of numerical data the leading significant digits are not uniformly distributed as one might suspect at first. An elegant way to explain and extend Benford’s law is to consider a random variable X > 0 and its expansion with integer base b ≥ 2. Hill (1995) stated the problem of finding distributions satisfying Benford’s law exactly. The special case of exponentially distributed random variables was studied by Engel and Leuenberger (2003): Such random variables satisfy the first digit law only approximatively, but precise estimates can be given; see Miller and Nigrini (2006) for an alternative proof and extensions. Sup P(U ∈ B) − Leb(B) → 0 as σ → ∞ This particular and similar results are typically derived via Fourier methods; see, for instance, Pinkham (1961) or Kontorovich and Miller (2005). We show that in case of Y being normally distributed with variance one or more, the distribution of the remainder U is very close to the uniform distribution on [0, 1)
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