Benford's law from 1881 to 2006

Benford or not Benford: a systematic but not always well-founded use of an elegant law in experimental fields

In this paper, we will see that the proportion of d as leading digit, d $\in$ 1, 9, in data (obtained thanks to the hereunder developed model) is more likely to follow a law whose probability distribution is determined by a specific upper bound, rather than Benford's Law. These probability distributions fluctuate around Benford's value as can often be observed in the literature in many naturally occurring collections of data (where the physical , biological or economical quantities considered are upper bounded). Knowing beforehand the value of the upper bound can be a way to find a better adju…

Benford's Law Beyond Independence: Tracking Benford Behavior in Copula Models

Benford's law describes a common phenomenon among many naturally occurring data sets and distributions in which the leading digits of the data are distributed with the probability of a first digit of $d$ base $B$ being $\log_{B}{\frac{d+1}{d}}$. As it often successfully detects fraud in medical trials, voting, science and finance, significant effort has been made to understand when and how distributions exhibit Benford behavior. Most of the previous work has been restricted to cases of independent variables, and little is known about situations involving dependence. We use copulas to investiga…

Fundamental Flaws in Feller's Classical Derivation of Benford's Law

Feller's classic text 'An Introduction to Probability Theory and its Applications' contains a derivation of the well known significant-digit law called Benford's law. More specifically, Feller gives a sufficient condition ("large spread") for a random variable $X$ to be approximately Benford distributed, that is, for $\log_{10}X$ to be approximately uniformly distributed modulo one. This note shows that the large-spread derivation, which continues to be widely cited and used, contains serious basic errors. Concrete examples and a new inequality clearly demonstrate that large spread (or large s…

Chains of distributions, hierarchical Bayesian models and Benford's Law

Kossovsky recently conjectured that the distribution of leading digits of a chain of probability distributions converges to Benford's law as the length of the chain grows. We prove his conjecture in many cases, and provide an interpretation in terms of products of independent random variables and a central limit theorem. An interesting consequence is that in hierarchical Bayesian models priors tend to satisfy Benford's Law as the number of levels of the hierarchy increases, which allows us to develop some simple tests (based on Benford's law) to test proposed models. We give explicit formulas …

Arithmetical Tugs of War and Benford's Law

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. The two essential prerequisites for data configuration with regards to compliance with Benford's Law are high order of magnitude and positive skewness with a tail falling to the right of the histogram, so that quantitative configuration is such that the small is numerous and the big is rare. A related topic in the study …

Random Consolidations and Fragmentations Cycles Lead to Benford' Law

Benford's Law predicts that the first significant digit on the leftmost side of numbers in real-life data is proportioned between all possible 1 to 9 digits approximately as in LOG(1 + 1/digit), so that low digits occur much more frequently than high digits in the first place. For example, digit 1 occurs approximately 30.1% in the first place in random numbers, while digit 9 occurs only approximately 4.6%. In this article it is shown that a process where a large enough set of identical quantities constantly alternates between minuscule random consolidations (summing two randomly chosen values …

First digit law from Laplace transform

The occurrence of digits 1 through 9 as the leftmost nonzero digit of numbers from real-world sources is distributed unevenly according to an empirical law, known as Benford's law or the first digit law. It remains obscure why a variety of data sets generated from quite different dynamics obey this particular law. We perform a study of Benford's law from the application of the Laplace transform, and find that the logarithmic Laplace spectrum of the digital indicator function can be approximately taken as a constant. This particular constant, being exactly the Benford term, explains the prevale…

Exponential Growth Series and Benford's Law

Exponential growth occurs when the growth rate of a given quantity is proportional to the quantity's current value. Surprisingly, when exponential growth data is plotted as a simple histogram disregarding the time dimension, a remarkable fit to the positively skewed k/x distribution is found, where the small is numerous and the big is rare. Such quantitative preference for the small has a corresponding digital preference known as Benford's Law which predicts that the first significant digit on the left-most side of numbers in typical real-life data is proportioned between all possible 1 to 9 d…

本福特定律的一个简洁证明

A concise proof of Benford's law

本文针对具有黎曼可积概率密度函数的分布，给出了本福特定律的简洁证明，并提出判断分布是否服从该定律的准则，为欺诈检测提供便利。

The first digit frequencies of primes and Riemann zeta zeros tend to uniformity following a size-dependent generalized Benford's law

Prime numbers seem to distribute among the natural numbers with no other law than that of chance, however its global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated sci- entists of all ages to search for local and global patterns in this distribution that eventually could shed light into the ultimate nature of primes. In this work we show that a generalization of the well known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in data sets, describes with astonishing precision the sta…

Benford定律及其推广的数学分析

A Mathematical Analysis of Benford's Law and its Generalization

本文数学分析Kossovsky推广的Benford定律，通过正态与反射Gumbel模型解释自然数据首位数字分布，并发现其可用于改进密度估计。

Towards A Better Understanding Of The Leading Digits Phenomena

That the logarithmic distribution manifests itself in the random as well as in the deterministic (multiplication processes) has long intrigued researchers in Benford's Law. In this article it is argued that it springs from one common intrinsic feature of their density curves. On the other hand, the profound dichotomy between the random and the deterministic in the context of Benford's Law is noted here, acknowledging the need to distinguish between them. From its very inception, the field has been suffering from a profound confusion and mixing of these two very different logarithmic flavors, c…

基于本福特定律的离散分数操纵测试

Manipulation testing based on Benford's Law for discrete scores

研究回归断点设计中变量操纵问题，基于本福特定律检测结构失衡，消除影响结果的参数，深入分析密度行为，能找出偏差源头与隐藏操纵，引入创新带宽选择方法及构建互补测试，实证证明该框架有增强保护价值。