[Paper Review] n-digit Benford distributed random variables

Azar Khosravani, Constantin Rasinariu|arXiv (Cornell University)|Apr 30, 2013

Benford’s Law and Fraud Detection4 references3 citations

TL;DR

This paper introduces the concept of $n$-digit Benford distributed random variables—random variables whose first $n$ significant digits follow Benford's law, while digits beyond the $n$-th may not. It uses the mod 1 map of the logarithmic density function $g^\dagger$ to characterize digit distributions, showing that uniform $g^\dagger$ implies scale- and base-invariance. The key contribution is a general construction method for $n$-digit Benford variables, illustrated with a 1-digit example via Mathematica simulation showing close agreement with theoretical Benford probabilities.

ABSTRACT

The scope of this paper is twofold. First, to emphasize the use of the mod 1 map in exploring the digit distribution of random variables. We show that the well-known base- and scale-invariance of Benford variables are consequences of their associated mod 1 density functions being uniformly distributed. Second, to introduce a new concept of the $n$-digit Benford variable. Such a variable is Benford in the first $n$ digits, but it is not guaranteed to have a logarithmic distribution beyond the $n$-th digit. We conclude the paper by giving a general construction method for $n$-digit Benford variables, and provide a concrete example.

Motivation & Objective

• To clarify the role of the mod 1 map in characterizing digit distributions of random variables.
• To demonstrate that scale-invariance and base-invariance in Benford-distributed variables stem from uniformity of the mod 1 density $g^\dagger$.
• To introduce and formalize the concept of $n$-digit Benford distributed random variables, which are Benford in the first $n$ digits but not necessarily beyond.
• To provide a general construction method for $n$-digit Benford variables using piecewise-defined mod 1 densities.
• To validate the construction with a numerical simulation of a 1-digit Benford variable using 100,000 data points in Mathematica.

Proposed method

• The mod 1 map is applied to the logarithmic density $g$ of a random variable $Y$ to define $g^\dagger(x) = \sum_{k=-\infty}^{\infty} g(x+k)$ for $x \in [0,1)$, which governs the distribution of significant digits.
• The probability of a specific sequence of first $n$ digits is computed as the integral of $g^\dagger$ over a logarithmic interval corresponding to that digit sequence.
• A general construction method is proposed by partitioning $[0,1)$ into $10^n$ intervals and defining $g^\dagger$ piecewise as normalized probability density functions over each interval to satisfy $n$-digit Benford conditions.
• For the 1-digit example, a non-uniform $g^\dagger$ is constructed using sine functions over intervals $[\log k, \log(k+1))$ for $k=1$ to $9$, ensuring correct first-digit probabilities.
• A 100,000-point discrete approximation of the 1-digit Benford variable is generated using Mathematica, with the density function $f(y)$ derived from $g^\dagger$ via $f(y) = g(\log y)/(y \ln 10)$.
• The simulated data are compared to theoretical Benford probabilities using histograms and statistical tables, confirming high accuracy.

Experimental results

Research questions

• RQ1 How does the mod 1 map of the logarithmic density function $g^\dagger$ determine the digit distribution of a random variable?
• RQ2 Why do scale-invariance and base-invariance in Benford-distributed variables arise from the uniformity of $g^\dagger$?
• RQ3 Can a random variable be constructed such that its first $n$ significant digits follow Benford's law, even if later digits do not?
• RQ4 What is a general method to construct $n$-digit Benford distributed random variables using piecewise-defined mod 1 densities?
• RQ5 How accurately can a 1-digit Benford variable be simulated numerically, and does the simulation match theoretical expectations?

Key findings

• A random variable is Benford in its first $n$ digits if and only if the integral of its mod 1 density $g^\dagger$ over the interval corresponding to a digit sequence equals $\log\left(1 + \frac{1}{10^{n-1}d_1 + \cdots + d_n}\right)$.
• The construction of $n$-digit Benford variables is achieved by partitioning $[0,1)$ into $10^n$ intervals and defining $g^\dagger$ as normalized densities over each interval, ensuring correct cumulative probabilities for all $n$-digit combinations.
• A 1-digit Benford variable was successfully simulated using 100,000 data points in Mathematica, with the simulated first-digit distribution closely matching theoretical Benford probabilities.
• The simulated probabilities for digits 1 through 9 were 0.3006, 0.1775, 0.1258, 0.0957, 0.0783, 0.0671, 0.0572, 0.0516, and 0.0463, respectively, compared to theoretical values of 0.3010, 0.1761, 0.1249, 0.0969, 0.0792, 0.0669, 0.0580, 0.0512, and 0.0458.
• The example with non-uniform $g^\dagger$ (e.g., sine-based) shows that while it produces a 1-digit Benford variable, it is not scale- or base-invariant, confirming that uniform $g^\dagger$ is necessary for such invariance.
• The paper establishes that $g^\dagger = 1$ (uniform) is both necessary and sufficient for scale- and base-invariance, and that this uniformity directly implies Benford’s law for the first digit.