On the significands of uniform random variables

2018 ◽  
Vol 55 (2) ◽  
pp. 353-367 ◽  
Author(s):  
Arno Berger ◽  
Isaac Twelves
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Benford’S Law ◽  
Elementary Calculation ◽  
Sharp Bounds ◽  
Large Spread ◽  
Benford's Law ◽  
Logarithmic Distribution

Abstract For all α > 0 and real random variables X, we establish sharp bounds for the smallest and the largest deviation of αX from the logarithmic distribution also known as Benford's law. In the case of uniform X, the value of the smallest possible deviation is determined explicitly. Our elementary calculation puts into perspective the recurring claims that a random variable conforms to Benford's law, at least approximately, whenever it has large spread.

Real-valued Random Processes

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Brownian Motion ◽  
Random Variables ◽  
Random Variable ◽  
Random Processes ◽  
Geometric Brownian Motion ◽  
Independent Random Variables ◽  
Benford’S Law ◽  
Random Maps ◽  
Random Samples ◽  
Benford's Law

Benford's law arises naturally in a variety of stochastic settings, including products of independent random variables, mixtures of random samples from different distributions, and iterations of random maps. This chapter provides the concepts and tools to analyze significant digits and significands for these basic random processes. Benford's law also arises in many other important fields of stochastics, such as geometric Brownian motion, random matrices, and Bayesian models, and the chapter may serve as a preparation for specialized literature on these advanced topics. By Theorem 4.2 a random variable X is Benford if and only if log ¦X¦ is uniformly distributed modulo one.

scholarly journals FIRST DIGIT DISTRIBUTION OF HADRON FULL WIDTH

2009 ◽  
Vol 24 (40) ◽  
pp. 3275-3282 ◽  
Author(s):  
LIJING SHAO ◽  
BO-QIANG MA
Keyword(s):  
Real World ◽  
Particle Physics ◽  
Benford’S Law ◽  
Full Width ◽  
Scale Invariant ◽  
Benford's Law ◽  
Science Domain ◽  
Logarithmic Distribution ◽  
First Time ◽  
Power Invariant

A phenomenological law, called Benford's law, states that the occurrence of the first digit, i.e. 1, 2,…, 9, of numbers from many real world sources is not uniformly distributed, but instead favors smaller ones according to a logarithmic distribution. We investigate, for the first time, the first digit distribution of the full widths of mesons and baryons in the well-defined science domain of particle physics systematically, and find that they agree excellently with the Benford distribution. We also discuss several general properties of Benford's law, i.e. the law is scale-invariant, base-invariant and power-invariant. This means that the lifetimes of hadrons also follow Benford's law.

Fourier Analysis and Benfordʼs Law

Benford's Law ◽  
2015 ◽  
Author(s):  
Steven J. Miller
Keyword(s):  
Distribution Function ◽  
Fourier Analysis ◽  
Random Matrix ◽  
Matrix Theory ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Benford’S Law ◽  
Good System ◽  
Benford's Law ◽  
Geometric Brownian Motions

This chapter continues the development of the theory of Benford's law. It uses Fourier analysis (in particular, Poisson Summation) to prove many systems either satisfy or almost satisfy the Fundamental Equivalence, and hence either obey Benford's law, or are well approximated by it. Examples range from geometric Brownian motions to random matrix theory to products and chains of random variables to special distributions. The chapter furthermore develops the notion of a Benford-good system. Unfortunately one of the conditions here concerns the cancelation in sums of translated errors related to the cumulative distribution function, and proving the required cancelation often requires techniques specific to the system of interest.

An Introduction to Benford's Law

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Random Variables ◽  
Benford’S Law ◽  
Late Nineteenth Century ◽  
Comprehensive Treatment ◽  
Open Problems ◽  
Significant Digit ◽  
Wide Range ◽  
Benford's Law ◽  
Primary Reference ◽  
Basic Facts

This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications. The book begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law. Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The book includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. The book can serve as both a primary reference and a basis for seminars and courses.

Introduction

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Numerical Data ◽  
Benford’S Law ◽  
Decimal Digit ◽  
Mathematical Framework ◽  
Benford's Law ◽  
History Of ◽  
Logarithmic Distribution ◽  
Benford Law ◽  
Special Case

This introductory chapter provides an overview of Benford' law. Benford's law, also known as the First-digit or Significant-digit law, is the empirical gem of statistical folklore that in many naturally occurring tables of numerical data, the significant digits are not uniformly distributed as might be expected, but instead follow a particular logarithmic distribution. In its most common formulation, the special case of the first significant (i.e., first non-zero) decimal digit, Benford's law asserts that the leading digit is not equally likely to be any one of the nine possible digits 1, 2, … , 9, but is 1 more than 30 percent of the time, and is 9 less than 5 percent of the time, with the probabilities decreasing monotonically in between. The remainder of the chapter covers the history of Benford' law, empirical evidence, early explanations and mathematical framework of Benford' law.

scholarly journals Benford’s law and continuous dependent random variables

2018 ◽  
Vol 388 ◽  
pp. 350-381 ◽  
Author(s):  
Thealexa Becker ◽  
David Burt ◽  
Taylor C. Corcoran ◽  
Alec Greaves-Tunnell ◽  
Joseph R. Iafrate ◽  
...  
Keyword(s):  
Random Variables ◽  
Benford’S Law ◽  
Dependent Random Variables ◽  
Benford's Law

scholarly journals LEI DE NEWCOMB BENFORD E AUDITORIA CONTÁBIL: UMA REVISÃO SISTEMÁTICA DE LITERATURA

2020 ◽  
Vol 17 (2) ◽  
pp. 111
Author(s):  
Caroline De Oliveira Orth ◽  
Anna Tamires Michaelsen ◽  
Arthur Frederico Lerner
Keyword(s):  
Systematic Review ◽  
Literature Review ◽  
Corporate Finance ◽  
Natural Number ◽  
Benford’S Law ◽  
Audit Tool ◽  
Research Gaps ◽  
Accounting Data ◽  
Benford's Law ◽  
Logarithmic Distribution

Lei de Newcomb Benford - LNB, foi concebida pelo astrônomo e matemático Simon Newcomb, em 1881. Seus estudos demonstraram que a ocorrência de um número natural, de modo espontâneo ou aleatório, não se dava na proporção esperada de 1/9, mas segundo uma distribuição logarítmica. Desde então, esta lei vem sendo testada em muitas áreas do conhecimento. Em finanças corporativas, os estudiosos têm testado a lei para investigar fraudes em dados contábeis. Contudo, ainda não há consenso sobre a eficácia da LNB nesse âmbito. Assim, o objetivo deste artigo é identificar os argumentos favoráveis e contrários, bem como os métodos de pesquisa e os principais achados das pesquisas sobre a aplicação da LNB como ferramenta de auditoria. Para tanto, aplicou-se uma Revisão Sistemática de Literatura, seguindo os passos de Levy e Ellis (2006). Deste modo, além de informações sobre autoria, modelos utilizados pelos autores para suportar suas conclusões e seus principais achados, apresentam-se lacunas de pesquisa, e as implicações para o futuro da pesquisa são discutidas.Palavras-chave: Lei de Newcomb Benford. Revisão sistemática. Auditoria contábil.ABSTRACTNewcomb Benford’s Law - LNB, was conceived by the astronomer and mathematician Simon Newcomb, in 1881. His studies showed that the occurrence of a natural number, spontaneously or randomly, did not occur in the expected proportion of 1/9, but according to a logarithmic distribution. Since then, this law has been tested in many areas of knowledge. In corporate finance, scholars have tested the law to investigate fraud in accounting data. However, there is still no consensus on the effectiveness of LNB in this area. Thus, the objective of this article is to identify the arguments for and against, as well as the research methods and the main findings of research on the application of LNB as an audit tool. For that, a Systematic Literature Review was applied, following the steps of Levy and Ellis (2006). Thus, in addition to information on authorship, models used by the authors to support their conclusions and main findings, research gaps are presented, and the implications for the future of research are discussed.Keywords: Newcomb Benford’s law. Systematic review. Accounting audit..

The Uniform Distribution and Benford's Law

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Uniform Distribution ◽  
Mathematical Theory ◽  
Random Variables ◽  
Benford’S Law ◽  
Benford's Law ◽  
Distribution Modulo One ◽  
Uniform Distribution Of Sequences ◽  
Uniform Distribution Modulo One ◽  
Distribution Modulo

The uniform distribution characterization of Benford's law is the most basic and powerful of all characterizations, largely because the mathematical theory of uniform distribution modulo one is very well developed for authoritative surveys. This chapter records and develops tools from that theory which will be used throughout this book to establish Benford behavior of sequences, functions, and random variables. Topics discussed include uniform distribution characterization of Benford's law, uniform distribution of sequences and functions, and uniform distribution of random variables.

The Benford Property

2015 ◽  
Author(s):  
Arno Berger ◽  
Theodore P. Hill
Keyword(s):  
Probability Distributions ◽  
Random Variables ◽  
Benford’S Law ◽  
Real Numbers ◽  
Benford's Law

In order to translate the informal versions of Benford's law into more precise formal statements, it is necessary to specify exactly what the Benford property means in various mathematical contexts. For the purpose of this book, the objects of interest fall mainly into three categories: sequences of real numbers, real-valued functions defined on [0,+ ∞), and probability distributions and random variables. This chapter defines Benford sequences, functions, and random variables, with examples of each.

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