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 the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford’s law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford’s law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book.
Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford’s law can serve as a productive meeting ground for researchers and practitioners in diverse fields.
Author(s): Steven J. Miller
Edition: 1
Publisher: Princeton University Press
Year: 2015
Language: English
Pages: 464
Tags: Benford Law
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 Law
3.1 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 Preliminaries
5.3 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 Accounting
7.1 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
Chapter 11. Measuring the Quality of European Statistics
11.1 Introduction
11.2 Macroeconomic Statistics in the EU
11.3 Benford’s Law and Macroeconomic Data
11.4 Conclusion
Chapter 12. Benford's Law and Fraud in Economic Research
12.1 Introduction
12.2 On Benford’s Law
12.3 Benford’s Law in Macroeconomic Data and Forecasts
12.4 Benford’s Law in Published Economic Research
12.5 Replication and Benford’s Law
12.6 Conclusions
Chapter 13. Testing for Strategic Manipulation of Economic and Financial Data
13.1 Benford in Economics
13.2 An Application to Value-at-Risk Data
PART V. APPLICATIONS III: SCIENCES
Chapter 14. Psychology and Benford's Law
14.1 A Behavioral Approach
14.2 Early Behavioral Research
14.3 Recent Research
14.4 Why Do People Approximate Benford’s Law?
14.5 Conclusions and Future Directions
Chapter 15. Managing Risk in Numbers Games: Benford's Law and the Small-Number Phenomenon
15.1 Introduction
15.2 Patterns in Number Selection: The Small-Number Phenomenon
15.3 Modeling Number Selection with Benford’s Law
15.4 Managerial Implications
15.5 Conclusions
Chapter 16. Benford's Law in the Natural Sciences
16.1 Introduction
16.2 Origins of Benford’s Law in Scientific Data
16.3 Examples of Benford’s Law in Scientific Data Sets
16.4 Applications of Benford’s Law in the Natural Sciences
16.5 Conclusion
Chapter 17. Generalizing Benford's Law: A Reexamination of Falsified Clinical Data
17.1 Introduction
17.2 Connecting Benford’s Law to Stigler’s Distribution
17.3 Connecting Stigler’s Law to Information-Theoretic Methods
17.4 Clinical Data
17.5 Summary and Implications
PART VI. APPLICATIONS IV: IMAGES
Chapter 18. Partial Volume Modeling of Medical Imaging Systems Using the Benford Distribution
18.1 Introduction
18.2 The Partial Volume Effect
18.3 Modeling of the PV Effect
18.4 Materials and Methods
18.5 Results and Discussion
18.6 Conclusions
Chapter 19. Application of Benford's Law to Images
19.1 Introduction
19.2 Background
19.3 Application of Benford’s Law to Images
19.4 A Fourier-Series-Based Model
19.5 Results Concerning Ensembles of DCT Coefficients
19.6 Jolion’s Results Revisited
19.7 Image Forensics
19.8 Summary
19.9 Appendix
PART VII. EXERCISES
Chapter 20. Exercises
20.1 A Quick Introduction to Benford’s Law
20.2 A Short Introduction to the Mathematical Theory of Benford’s Law
20.3 Fourier Analysis and Benford’s Law
20.4 Benford’s Law Geometry
20.5 Explicit Error Bounds via Total Variation
20.6 Lévy Processes and Benford’s Law
20.7 Benford’s Law as a Bridge between Statistics and Accounting
20.8 Detecting Fraud and Errors Using Benford’s Law
20.9 Can Vote Counts’ Digits and Benford’s Law Diagnose Elections?
20.10 Complementing Benford’s Law for Small N: A Local Bootstrap
20.11 Measuring the Quality of European Statistics
20.12 Benford’s Law and Fraud in Economic Research
20.13 Testing for Strategic Manipulation of Economic and Financial Data
20.14 Psychology and Benford’s Law
20.15 Managing Risk in Numbers Games: Benford’s Law and the Small-Number Phenomenon
20.16 Benford’s Law in the Natural Sciences
20.17 Generalizing Benford’s Law: A Reexamination of Falsified Clinical Data
20.18 Partial Volume Modeling of Medical Imaging Systems Using the Benford Distribution
20.19 Application of Benford’s Law to Images
Bibliography
Index
