Benford's Law: Theory and Applications

465 Pages·2015·13.154 MB·English

by  Steven J. Miller

Benfordʼs Law: Theory and
Applications

Benfordʼs Law: Theory and
Applications
Edited by Steven J. Miller
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright(cid:2)c 2015byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,41WilliamStreet,Princeton,NewJersey08540
IntheUnitedKingdom: PrincetonUniversityPress,6OxfordStreet, Woodstock,OxfordshireOX20
1TW
press.princeton.edu
AllRightsReserved
ISBN978-0-691-14761-1
BritishLibraryCataloging-in-PublicationDataisavailable
ThisbookhasbeencomposedinTimesRomaninLATEX
Thepublisher wouldliketoacknowledge theauthors ofthisvolumeforprovidingthecamera-ready
copyfromwhichthisbookwasprinted.
Printedonacid-freepaper.∞
PrintedintheUnitedStatesofAmerica
10987654321
Dedicatedtomycolleaguesandstudentsformanyfunyearsexploring
Benford’sLawtogether,tomyparentsArleneandWilliamfortheirsup-
portandencouragementovertheyears,andtothenumber1forbeing
suchagood,frequentcompanion. — SJM,Williamstown,MA,2015.

Contents
Foreword xiii
Preface xvii
Notation xxiii
PARTI. GENERALTHEORYI:BASISOFBENFORDʼSLAW 1
Chapter1. AQuickIntroductiontoBenfordʼsLaw 3
1.1 Overview 3
1.2 Newcomb 4
1.3 Benford 5
1.4 StatementofBenford’sLaw 7
1.5 ExamplesandExplanations 8
1.6 Questions 16
Chapter2. AShortIntroductiontotheMathematicalTheoryofBenfordʼs
Law 23
2.1 Introduction 23
2.2 SignificantDigitsandtheSignificand 24
2.3 TheBenfordProperty 28
2.4 CharacterizationsofBenford’sLaw 31
2.5 Benford’sLawforDeterministicProcesses 43
2.6 Benford’sLawforRandomProcesses 55
Chapter3. FourierAnalysisandBenfordʼsLaw 68
3.1 Introduction 68
3.2 Benford-GoodProcesses 70
3.3 ProductsofIndependentRandomVariables 81
3.4 ChainsofRandomVariables 88
3.5 WeibullRandomVariables,SurvivalDistributions,andOrderStatistics 96
3.6 BenfordnessofCauchyDistributions 102
PART II. GENERAL THEORY II: DISTRIBUTIONS AND RATES OF
CONVERGENCE 107
Chapter4. BenfordʼsLawGeometry 109
viii CONTENTS
4.1 Introduction 109
4.2 CommonProbabilityDistributions 111
4.3 ProbabilityDistributionsSatisfyingBenford’sLaw 113
4.4 Conclusions 118
Chapter5. ExplicitErrorBoundsviaTotalVariation 119
5.1 Introduction 119
5.2 Preliminaries 120
5.3 ErrorBoundsinTermsofTV(f) 123
5.4 ErrorBoundsinTermsofTV(f(k)) 125
5.5 Proofs 130
Chapter6. LévyProcessesandBenfordʼsLaw 135
6.1 Overview,BasicDefinitions,andExamples 136
6.2 ExpectationsofNormalizedFunctionals 149
6.3 A.S.ConvergenceofNormalizedFunctionals 155
6.4 NecessaryandSufficientConditionsfor(D)or(SC) 161
6.5 StatisticalApplications 164
6.6 Appendix1:AnotherVariantofPoissonSummation 169
6.7 Appendix2:AnElementaryPropertyofConditionalExpectations 172
PARTIII. APPLICATIONSI:ACCOUNTINGANDVOTEFRAUD 175
Chapter7. BenfordʼsLawasaBridgebetweenStatisticsandAccounting 177
7.1 TheCaseforAccountantstoLearnStatistics 177
7.2 TheFinancialStatementAuditor’sWorkEnvironment 179
7.3 PracticalandStatisticalHypotheses 183
7.4 FromStatisticalHypothesistoDecisionMaking 185
7.5 ExampleforClassroomUse 188
7.6 ConclusionandRecommendations 189
Chapter8. DetectingFraudandErrorsUsingBenfordʼsLaw 191
8.1 Introduction 191
8.2 Benford’sOriginalPaper 192
8.3 CaseStudieswithAuthenticData 193
8.4 CaseStudieswithFraudulentData 202
8.5 Discussion 210
Chapter 9. Can Vote Countsʼ Digits and Benfordʼs Law Diagnose Elec-
tions? 212
9.1 Introduction 212
9.2 2BLandPrecinctVoteCounts 213
9.3 AnExampleofStrategicBehaviorbyVoters 218
9.4 Discussion 222
Chapter 10. Complementing Benfordʼs Law for Small N: A Local Boot-
strap 223
10.1 The2009IranianPresidentialElection 223
CONTENTS ix
10.2 ApplicabilityofBenford’sLawandtheK7Anomaly 224
10.3 AConservativeAlternativetoBenford’sLaw:ASmallN,Empirical,Local
BootstrapModel 227
10.4 UsingaSuspectedAnomalytoSelectSubsetsoftheData 229
10.5 WhenLocalBootstrapsComplementBenford’sLaw 231
PARTIV. APPLICATIONSII:ECONOMICS 233
Chapter11. MeasuringtheQualityofEuropeanStatistics 235
11.1 Introduction 235
11.2 MacroeconomicStatisticsintheEU 236
11.3 Benford’sLawandMacroeconomicData 237
11.4 Conclusion 242
Chapter12. BenfordʼsLawandFraudinEconomicResearch 244
12.1 Introduction 244
12.2 OnBenford’sLaw 245
12.3 Benford’sLawinMacroeconomicDataandForecasts 248
12.4 Benford’sLawinPublishedEconomicResearch 250
12.5 ReplicationandBenford’sLaw 253
12.6 Conclusions 255
Chapter13. TestingforStrategicManipulationofEconomicandFinancial
Data 257
13.1 BenfordinEconomics 257
13.2 AnApplicationtoValue-at-RiskData 260
PARTV. APPLICATIONSIII:SCIENCES 265
Chapter14. PsychologyandBenfordʼsLaw 267
14.1 ABehavioralApproach 267
14.2 EarlyBehavioralResearch 268
14.3 RecentResearch 270
14.4 WhyDoPeopleApproximateBenford’sLaw? 273
14.5 ConclusionsandFutureDirections 274
Chapter15. ManagingRiskinNumbersGames: BenfordʼsLawandthe
Small-NumberPhenomenon 276
15.1 Introduction 276
15.2 PatternsinNumberSelection:TheSmall-NumberPhenomenon 277
15.3 ModelingNumberSelectionwithBenford’sLaw 280
15.4 ManagerialImplications 284
15.5 Conclusions 289
Chapter16. BenfordʼsLawintheNaturalSciences 290
16.1 Introduction 290
16.2 OriginsofBenford’sLawinScientificData 291