A Widespread Error in the Use of Benford's Law to Detect Election and Other Fraud

The Inverse Gamma Distribution and Benford's Law

According to Benford's Law, many data sets have a bias towards lower leading digits (about $30\%$ are $1$'s). The applications of Benford's Law vary: from detecting tax, voter and image fraud to determining the possibility of match-fixing in competitive sports. There are many common distributions that exhibit such bias, i.e. they are almost Benford. These include the exponential and the Weibull distributions. Motivated by these examples and the fact that the underlying distribution of factors in protein structure follows an inverse gamma distribution, we determine the closeness of this distrib…

高维碎裂过程的 Benford 行为

Benford Behavior of a Higher-Dimensional Fragmentation Process

本文将一维棍棒碎裂长度服从 Benford 定律的结果推广到任意有限维体积，并猜想低维体积在无限制碎裂下同样遵循该定律。

The Weibull Distribution and Benford's Law

Benford's law states that many data sets have a bias towards lower leading digits (about $30\%$ are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford's law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by …

Benford's Law and Continuous Dependent Random Variables

Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This phenomenon, known as Benford's Law, has many applications, ranging from detecting tax fraud for the IRS to analyzing round-off errors in computer science. The central question is determining which data sets follow Benford's law. Inspired by natural processes such as particle decay, our work examines models for the decomposition of conserved quantities. We prove t…

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。

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…