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# Benford's Law : Theory and Applications

Material type: TextSeries: eBooks on Demand.Publisher: Princeton : Princeton University Press, 2015Description: 1 online resource (465 p.).ISBN: 9781400866595.Genre/Form: Electronic books.Additional physical formats: Print version:: Benford's Law : Theory and ApplicationsDDC classification: 519.2/4 Online resources: Click here to view this ebook.
Contents:
Cover; Title ; Copyright ; Dedication; Contents; Foreword; Preface; Notation; PART I. GENERAL THEORY I: BASIS OF BENFORD'S LAW; Chapter 1. A Quick Introduction to Benford's Law; 1.1 Overview; 1.2 Newcomb; 1.3 Benford; 1.4 Statement of Benford's Law; 1.5 Examples and Explanations; 1.6 Questions; Chapter 2. A Short Introduction to the Mathematical Theory of Benford's Law; 2.1 Introduction; 2.2 Significant Digits and the Significand; 2.3 The Benford Property; 2.4 Characterizations of Benford's Law; 2.5 Benford's Law for Deterministic Processes; 2.6 Benford's Law for Random Processes
Chapter 3. Fourier Analysis and Benford's Law3.1 Introduction; 3.2 Benford-Good Processes; 3.3 Products of Independent Random Variables; 3.4 Chains of Random Variables; 3.5 Weibull Random Variables, Survival Distributions, and Order Statistics; 3.6 Benfordness of Cauchy Distributions; PART II. GENERAL THEORY II: DISTRIBUTIONS AND RATES OF CONVERGENCE; Chapter 4. Benford's Law Geometry; 4.1 Introduction; 4.2 Common Probability Distributions; 4.3 Probability Distributions Satisfying Benford's Law; 4.4 Conclusions; Chapter 5. Explicit Error Bounds via Total Variation; 5.1 Introduction
5.2 Preliminaries5.3 Error Bounds in Terms of TV(f); 5.4 Error Bounds in Terms of TV(f(k)); 5.5 Proofs; Chapter 6. Lévy Processes and Benford's Law; 6.1 Overview, Basic Definitions, and Examples; 6.2 Expectations of Normalized Functionals; 6.3 A.S. Convergence of Normalized Functionals; 6.4 Necessary and Sufficient Conditions for (D) or (SC); 6.5 Statistical Applications; 6.6 Appendix 1: Another Variant of Poisson Summation; 6.7 Appendix 2: An Elementary Property of Conditional Expectations; PART III. APPLICATIONS I: ACCOUNTING AND VOTE FRAUD
Chapter 7. Benford's Law as a Bridge between Statistics and Accounting7.1 The Case for Accountants to Learn Statistics; 7.2 The Financial Statement Auditor's Work Environment; 7.3 Practical and Statistical Hypotheses; 7.4 From Statistical Hypothesis to Decision Making; 7.5 Example for Classroom Use; 7.6 Conclusion and Recommendations; Chapter 8. Detecting Fraud and Errors Using Benford's Law; 8.1 Introduction; 8.2 Benford's Original Paper; 8.3 Case Studies with Authentic Data; 8.4 Case Studies with Fraudulent Data; 8.5 Discussion
Chapter 9. Can Vote Counts' Digits and Benford's Law Diagnose Elections?9.1 Introduction; 9.2 2BL and Precinct Vote Counts; 9.3 An Example of Strategic Behavior by Voters; 9.4 Discussion; Chapter 10. Complementing Benford's Law for Small N: A Local Bootstrap; 10.1 The 2009 Iranian Presidential Election; 10.2 Applicability of Benford's Law and the K7 Anomaly; 10.3 A Conservative Alternative to Benford's Law: A Small N, Empirical, Local Bootstrap Model; 10.4 Using a Suspected Anomaly to Select Subsets of the Data; 10.5 When Local Bootstraps Complement Benford's Law
PART IV. APPLICATIONS II: ECONOMICS
Summary: Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain t
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Cover; Title ; Copyright ; Dedication; Contents; Foreword; Preface; Notation; PART I. GENERAL THEORY I: BASIS OF BENFORD'S LAW; Chapter 1. A Quick Introduction to Benford's Law; 1.1 Overview; 1.2 Newcomb; 1.3 Benford; 1.4 Statement of Benford's Law; 1.5 Examples and Explanations; 1.6 Questions; Chapter 2. A Short Introduction to the Mathematical Theory of Benford's Law; 2.1 Introduction; 2.2 Significant Digits and the Significand; 2.3 The Benford Property; 2.4 Characterizations of Benford's Law; 2.5 Benford's Law for Deterministic Processes; 2.6 Benford's Law for Random Processes

Chapter 3. Fourier Analysis and Benford's Law3.1 Introduction; 3.2 Benford-Good Processes; 3.3 Products of Independent Random Variables; 3.4 Chains of Random Variables; 3.5 Weibull Random Variables, Survival Distributions, and Order Statistics; 3.6 Benfordness of Cauchy Distributions; PART II. GENERAL THEORY II: DISTRIBUTIONS AND RATES OF CONVERGENCE; Chapter 4. Benford's Law Geometry; 4.1 Introduction; 4.2 Common Probability Distributions; 4.3 Probability Distributions Satisfying Benford's Law; 4.4 Conclusions; Chapter 5. Explicit Error Bounds via Total Variation; 5.1 Introduction

5.2 Preliminaries5.3 Error Bounds in Terms of TV(f); 5.4 Error Bounds in Terms of TV(f(k)); 5.5 Proofs; Chapter 6. Lévy Processes and Benford's Law; 6.1 Overview, Basic Definitions, and Examples; 6.2 Expectations of Normalized Functionals; 6.3 A.S. Convergence of Normalized Functionals; 6.4 Necessary and Sufficient Conditions for (D) or (SC); 6.5 Statistical Applications; 6.6 Appendix 1: Another Variant of Poisson Summation; 6.7 Appendix 2: An Elementary Property of Conditional Expectations; PART III. APPLICATIONS I: ACCOUNTING AND VOTE FRAUD

Chapter 7. Benford's Law as a Bridge between Statistics and Accounting7.1 The Case for Accountants to Learn Statistics; 7.2 The Financial Statement Auditor's Work Environment; 7.3 Practical and Statistical Hypotheses; 7.4 From Statistical Hypothesis to Decision Making; 7.5 Example for Classroom Use; 7.6 Conclusion and Recommendations; Chapter 8. Detecting Fraud and Errors Using Benford's Law; 8.1 Introduction; 8.2 Benford's Original Paper; 8.3 Case Studies with Authentic Data; 8.4 Case Studies with Fraudulent Data; 8.5 Discussion

Chapter 9. Can Vote Counts' Digits and Benford's Law Diagnose Elections?9.1 Introduction; 9.2 2BL and Precinct Vote Counts; 9.3 An Example of Strategic Behavior by Voters; 9.4 Discussion; Chapter 10. Complementing Benford's Law for Small N: A Local Bootstrap; 10.1 The 2009 Iranian Presidential Election; 10.2 Applicability of Benford's Law and the K7 Anomaly; 10.3 A Conservative Alternative to Benford's Law: A Small N, Empirical, Local Bootstrap Model; 10.4 Using a Suspected Anomaly to Select Subsets of the Data; 10.5 When Local Bootstraps Complement Benford's Law

PART IV. APPLICATIONS II: ECONOMICS

Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain t

Description based upon print version of record.

### Reviews provided by Syndetics

#### CHOICE Review

After a sleepy 135 years, Benford's Law has exploded into a hot topic, the subject of three monographs now, all since 2012. The law says that leading digits of quantities appearing in large data sets often show a counterintuitive but precisely predictable bias toward small values. Mathematicians study when and why by verifying the law in theoretical models (by diverse arguments), then fitting real-life data to appropriate models. One quarter of this book sets out mathematical basics--another treatment of this appears in Arno Berger and Theodore Hill's An Introduction to Benford's Law (2015). Since Benford's Law cannot hold exactly for finite data sets, real-world applications require convergence rate estimates; some chapters in the next quarter of the book address this advanced topic. The other half of the book develops applications. Sometimes failures of Benford's Law in data where it should otherwise apply signal fraud in scientific research, voting, economic surveys, or accounting. Benford's Law has ingenious applications to image analysis, helping detect surreptitious manipulation or embedded steganography (hidden codes). The law also provides a subtle tool for analyzing ambiguous regions in diagnostic medical imaging. The book concludes with a section of challenging exercises well suited to undergraduates. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, researchers, professionals. --David V. Feldman, University of New Hampshire

### Author notes provided by Syndetics

Steven J. Miller is associate professor of mathematics at Williams College. He is the coauthor of An Invitation to Modern Number Theory (Princeton).

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