Benford's Law: Theory and Applications

by Steven J. Miller (Editor)

Overview

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.

Product Details

ISBN-13:	9780691147611
Publisher:	Princeton University Press
Publication date:	05/26/2015
Edition description:	New Edition
Pages:	464
Product dimensions:	6.50(w) x 9.40(h) x 1.50(d)

About the Author

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

Table of Contents

Foreword xiii

Preface xvii

Notation xxiii

Part I General Theory I: Basis of Benford's Law 1

Chapter 1 A Quick Introduction to Benford's Law 3

1.1 Overview 3

1.2 Newcomb 4

1.3 Benford 5

1.4 Statement of Benford's Law 7

1.5 Examples and Explanations 8

1.6 Questions 16

Chapter 2 A Short Introduction to the Mathematical Theory of Benford's Law 23

2.1 Introduction 23

2.2 Significant Digits and the Significand 24

2.3 The Benford Property 28

2.4 Characterizations of Benford's Law 31

2.5 Benford's Law for Deterministic Processes 43

2.6 Benford's Law for Random Processes 55

Chapter 3 Fourier Analysis and Benford's Law 68

3.1 Introduction 68

3.2 Benford-Good Processes 70

3.3 Products of Independent Random Variables 81

3.4 Chains of Random Variables 88

3.5 Weibull Random Variables, Survival Distributions, and Order Statistics 96

3.6 Benfordness of Cauchy Distributions 102

Part II General Theory II: Distributions and Rates of Convergence 107

Chapter 4 Benford's Law Geometry 109

4.1 Introduction 109

4.2 Common Probability Distributions 111

4.3 Probability Distributions Satisfying Benford's Law 113

4.4 Conclusions 118

Chapter 5 Explicit Error Bounds via Total Variation 119

5.1 Introduction 119

5.2 Preliminaries 120

5.3 Error Bounds in Terms of TV(f) 123

5.4 Error Bounds in Terms of TV(f^(k)) 125

5.5 Proofs 130

Chapter 6 Levy Processes and Benford's Law 135

6.1 Overview, Basic Definitions, and Examples 136

6.2 Expectations of Normalized Functionals 149

6.3 A.S. Convergence of Normalized Functionals 155

6.4 Necessary and Sufficient Conditions for (D) or (SC) 161

6.5 Statistical Applications 164

6.6 Appendix 1: Another Variant of Poisson Summation 169

6.7 Appendix 2: An Elementary Property of Conditional Expectations 172

Part III Applications I: Accounting and Vote Fraud 175

Chapter 7 Benford's Law as a Bridge between Statistics and Accounting 177

7.1 The Case for Accountants to Learn Statistics 177

7.2 The Financial Statement Auditor's Work Environment 179

7.3 Practical and Statistical Hypotheses 183

7.4 From Statistical Hypothesis to Decision Making 185

7.5 Example for Classroom Use 188

7.6 Conclusion and Recommendations 189

Chapter 8 Detecting Fraud and Errors Using Benford's Law 191

8.1 Introduction 191

8.2 Benford's Original Paper 192

8.3 Case Studies with Authentic Data 193

5.1 Case Studies with Fraudulent Data 202

8.1 Discussion 210

Chapter 9 Can Vote Counts' Digits and Benford's Law Diagnose Elections? 212

9.1 Introduction 212

9.2 2BL and Precinct Vote Counts 213

9.3 An Example of Strategic Behavior by Voters 218

9.4 Discussion 222

Chapter 10 Complementing Benford's Law for Small N: A Local Bootstrap 223

10.1 The 2009 Iranian Presidential Election 223

10.2 Applicability of Benford's Law and the K7 Anomaly 224

10.3 A Conservative Alternative to Benford's Law: A Small N, Empirical, Local Bootstrap Model 227

10.4 Using a Suspected Anomaly to Select Subsets of the Data 229

10.5 When Local Bootstraps Complement Benford's Law 231

Part IV Applications II: Economics 233

Chapter 11 Measuring the Quality of European Statistics 235

11.1 Introduction 235

11.2 Macroeconomic Statistics in the EU 236

11.3 Benford's Law and Macroeconomic Data 237

11.4 Conclusion 242

Chapter 12 Benford's Law and Fraud in Economic Research 244

12.1 Introduction 244

12.2 On Benford's Law 245

12.3 Benford's Law in Macroeconomic Data and Forecasts 248

12.4 Benford's Law in Published Economic Research 250

12.5 Replication and Benford's Law 253

12.6 Conclusions 255

Chapter 13 Testing for Strategic Manipulation of Economic and Financial Data 257

13.1 Benford in Economics 257

13.2 An Application to Value-at-Risk Data 260

Part V Applications III: Sciences 265

Chapter 14 Psychology and Benford's Law 267

14.1 A Behavioral Approach 267

14.2 Early Behavioral Research 268

14.3 Recent Research 270

14.4 Why Do People Approximate Benford's Law? 273

14.5 Conclusions and Future Directions 274

Chapter 15 Managing Risk in Numbers Games: Benford's Law and the Small-Number Phenomenon 276

15.1 Introduction 276

15.2 Patterns in Number Selection: The Small-Number Phenomenon 277

15.3 Modeling Number Selection with Benford's Law 280

15.4 Managerial Implications 284

15.5 Conclusions 289

Chapter 16 Benford's Law in the Natural Sciences 290

16.1 Introduction 290

16.2 Origins of Benford's Law in Scientific Data 291

16.3 Examples of Bernard's Law in Scientific Data Sets 294

16.4 Applications of Benford's Law in the Natural Sciences 300

16.5 Conclusion 303

Chapter 17 Generalizing Benford's Law: A Reexamination of Falsified Clinical Data 304

17.1 Introduction 304

17.2 Connecting Benford's Law to Stigler's Distribution 305

17.3 Connecting Stigler's Law to Information-Theoretic Methods 307

17.4 Clinical Data 310

17.5 Summary and Implications 315

Part VI Applications IV: Images 317

Chapter 18 Partial Volume Modeling of Medical Imaging Systems Using the Benford Distribution 319

18.1 Introduction 319

18.2 The Partial Volume Effect 322

18.3 Modeling of the PV Effect 324

18.4 Materials and Methods 331

18.5 Results and Discussion 334

18.6 Conclusions 337

Chapter 19 Application of Benford's Law to Images 338

19.1 Introduction 338

19.2 Background 339

19.3 Application of Benford's Law to Images 340

19.4 AEourier-Series-Based Model 343

19.5 Results Concerning Ensembles of DCT Coefficients 350

19.6 Jolion's Results Revisited 354

19.7 Image Forensics 360

19.8 Summary 365

19.9 Appendix 366

Part VII Exercises 371

Chapter 20 Exercises 373

20.1 A Quick Introduction to Benford's Law 373

20.2 A Short Introduction to the Mathematical Theory of Benford's Law 376

20.3 Fourier Analysis and Benford's Law 377

20.4 Benford's Law Geometry 386

20.5 Explicit Error Bounds via Total Variation 386

20.6 Levy Processes and Benford's Law 387

20.7 Benford's Law as a Bridge between Statistics and Accounting 393

20.8 Detecting Fraud and Errors Using Benford's Law 395

20.9 Can Vote Counts' Digits and Benford's Law Diagnose Elections? 396

20.10 Complementing Benford's Law for Small N: A Local Bootstrap 396

20.11 Measuring the Quality of European Statistics 396

20.12 Benford's Law and Fraud in Economic Research 397

20.13 Testing for Strategic Manipulation of Economic and Financial Data 398

20.14 Psychology and Benford's Law 398

20.15 Managing Risk in Numbers Games: Benford's Law and the Small-Number Phenomenon 399

20.16 Benford's Law in the Natural Sciences 399

20.17 Generalizing Benford's Law: A Reexamination of Falsified Clinical Data 400

20.18 Partial Volume Modeling of Medical Imaging Systems Using the Benford Distribution 401

20.19 Application of Benford's Law to Images 401

Bibliography 402

Index 433

What People are Saying About This

From the Publisher

"This important, impressive, and well-crafted book presents the theory and many diverse applications of Benford's law. The contributors cover a rich and fascinating selection of topics that will appeal to people with an interest in mathematics and statistics as well as experts in a broad range of disciplines."—Iddo Ben-Ari, University of Connecticut

"Benford's law has been underappreciated by academics and scholars since its appearance, and a book of this kind has long been needed. It offers a well-organized, unified presentation of Benford's law from both a theoretical point of view and an applications standpoint."—Marco Corazza, Ca' Foscari University of Venice (Italy)

Editorial Reviews

"This book will prove to be both a valuable reference and a first source to turn to for whoever is interested in the mathematical genesis and empirical usefulness of Benford's law."—-Walter Krämer, Statistical Papers

From the Publisher