Related papers: Benford's Law and Continuous Dependent Random Vari…
We explain Kossovsky's generalization of Benford's law which is a formula that approximates the distribution of leftmost digits in finite sequences of natural data and apply it to six sequences of data including populations of US cities and…
The first digit (FD) phenomenon i.e., the significant digits of numbers in large data are often distributed according to a logarithmically decreasing function was first reported by S. Newcomb and then many decades later independently by F.…
This paper has several major purposes. The central purpose is to describe the "Benford analysis" of a positive random variable and to summarize some results from investigations into base dependence of Benford random variables. The principal…
Benford's law describes the distribution of the first digit of numbers appearing in a wide variety of numerical data, including tax records, and election outcomes, and has been used to raise "red flags" about potential anomalies in the data…
There are now many theoretical explanations for why Benford's law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in depth recurrence relations. As these are discrete…
Suppose you look at today's stock prices and bet on the value of the first digit. One could guess that a fair bet should correspond to the frequency of $1/9 = 11.11%$ for each digit from 1 to 9. This is by no means the case, and one can…
We study the concatenated Fibonacci constant $\mathcal{F} := 0.F_{1}F_{2}F_{3}\cdots = 0.11235813\cdots$, obtained by concatenating the Fibonacci numbers in the fractional part, and ask whether it is normal. We show that several classical…
The occurrence of digits one through nine as the leftmost nonzero digit of numbers from real world sources is often not uniformly distributed, but instead, is distributed according to a logarithmic law, known as Benford's law. Here, we…
Nonextensive statistics, characterized by a nonextensive parameter $q$, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore…
A simple method to derive parametric analytical extensions of Benford's law for first digits of numerical data is proposed. Two generalized Benford distributions are considered, namely the two-sided power Benford distribution and the new…
We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geometric law for codelength. Maximizing its entropy under a…
Considering the first significant digits (noted d) in data sets of dissipation for turbulent flows, the probability to find a given number (d=1 or 2 or... 9) would be 1/9 for an uniform distribution. Instead the probability closely follows…
It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable…
We provide conditions on dependent and on non-stationary random variables $X_n$ ensuring that the mantissa of the sequence of products $\left(\prod_{1}^{n}X_k\right)$ is almost surely distributed following the Benford's law or converges in…
Benford's law is often used as a support to critical decisions related to data quality or the presence of data manipulations or even fraud. However, many authors argue that conventional statistical tests will reject the null of data…
Benford's law is an empirical observation, first reported by Simon Newcomb in 1881 and then independently by Frank Benford in 1938: the first significant digits of numbers in large data are often distributed according to a logarithmically…
A recent article by Alexopoulos and Leontsinis presented empirical evidence that the first digits of the distances to galaxies are a reasonably good fit to the probabilities predicted by Benford's law, the well known logarithmic statistical…
In this paper, we present a possible theoretical explanation for benford's law. We develop a recursive relation between the probabilities, using simple intuitive ideas. We first use numerical solutions of this recursion and verify that the…
We study multi-digit correlations in Benford sequences b^n for integer bases 2 <= b <= 1000, measuring dependence via conditional mutual information (CMI). A resonance ratio derived from the continued fraction expansion of log_10(b)…
The goal of this note is to show that a widespread claim about Benford's Law, namely, that the range of every Benford distribution spans at least several orders of magnitude, is false. The proof is constructive and concrete examples are…