The Conway–Maxwell–Poisson Distribution

356 Pages·2023·25.263 MB·English

by  Kimberly F. Sellers

Monographs
The Conway-
Maxwell-
Poisson
Distribution
Kimberly F. Sellers

TheConway–Maxwell–PoissonDistribution
WhilethePoissondistributionisaclassicalstatisticalmodelforcountdata,thedistribu-
tionalmodelhingesontheconstrainingpropertythatitsmeanequalsitsvariance.This
textinsteadintroducestheConway–Maxwell–Poissondistributionandmotivatesitsuse
indevelopingflexiblestatisticalmethodsbasedonitsdistributionalform.
This two-parameter model not only contains the Poisson distribution as a special
casebut,initsabilitytoaccountfordataover-orunder-dispersion,encompassesboth
thegeometricandBernoullidistributions.Theresultingstatisticalmethodsserveina
multitudeofways,fromanexploratorydataanalysistooltoaflexiblemodelingimpetus
forvariedstatisticalmethodsinvolvingcountdata.
Thefirstcomprehensivereferenceonthesubject,thistextcontainsnumerousillus-
trativeexamplesdemonstratingRcodeandoutput.Itisessentialreadingforacademics
in statistics and data science, as well as quantitative researchers and data analysts in
economics,biostatisticsandotherapplieddisciplines.
KIMBERLY F. SELLERSisProfessorintheDepartmentofMathematicsandStatis-
tics at Georgetown Universityand aresearcher at the U.S. CensusBureau. Her work
hascontributedtocountdataresearchandsoftwareforthelast15years.SheisaFel-
lowoftheAmericanStatisticalAssociationandanElectedMemberoftheInternational
StatisticalInstitute.
INSTITUTE OF MATHEMATICAL STATISTICS
MONOGRAPHS
EditorialBoard
MarkHandcock,GeneralEditor(UCLA)
JohnAston(UniversityofCambridge)
ArnaudDoucet(UniversityofOxford)
RamonvanHandel(PrincetonUniversity)
IMS Monographs are concise research monographs of high quality on any branch of
statistics or probability of sufficient interest to warrant publication as books. Some
concern relatively traditional topics in need of up-to-date assessment. Others are on
emergingthemes.Inallcasestheobjectiveistoprovideabalancedviewofthefield.
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The Conway–Maxwell–Poisson
Distribution
KIMBERLY F. SELLERS
GeorgetownUniversity
ShaftesburyRoad,CambridgeCB28EA,UnitedKingdom
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CambridgeUniversityPressispartofCambridgeUniversityPress&Assessment,
adepartmentoftheUniversityofCambridge.
WesharetheUniversity’smissiontocontributetosocietythroughthepursuitof
education,learningandresearchatthehighestinternationallevelsofexcellence.
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108481106
DOI:10.1017/9781108646437
©KimberlyF.Sellers2023
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisions
ofrelevantcollectivelicensingagreements,noreproductionofanypartmaytake
placewithoutthewrittenpermissionofCambridgeUniversityPress&Assessment.
Firstpublished2023
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-1-108-48110-6Hardback
CambridgeUniversityPress&Assessmenthasnoresponsibilityforthepersistence
oraccuracyofURLsforexternalorthird-partyinternetwebsitesreferredtointhis
publicationanddoesnotguaranteethatanycontentonsuchwebsitesis,orwill
remain,accurateorappropriate.
Tothosethat“count”mostinmylife:
Myfamily,especiallymyson

Contents
ListofFigures pagexii
ListofTables xiv
Preface xxi
Acknowledgments xxiii
1 Introduction:CountDataContainingDispersion 1
1.1 PoissonDistribution 2
1.1.1 RComputing 4
1.2 DataOver-dispersion 5
1.2.1 RComputing 7
1.3 DataUnder-dispersion 10
1.3.1 RComputing 11
1.4 WeightedPoissonDistributions 16
1.5 Motivation,andSummaryoftheBook 18
2 TheConway–Maxwell–Poisson(COM–Poisson)Distribution 22
2.1 TheDerivation/Motivation:AFlexibleQueueingModel 23
2.2 TheProbabilityDistribution 25
2.2.1 RComputing 28
2.3 DistributionalandStatisticalProperties 35
2.3.1 RComputing 39
2.4 ParameterEstimationandStatisticalInference 40
2.4.1 CombiningCOM–PoissonnessPlotwithWeighted
LeastSquares 40
2.4.2 MaximumLikelihoodEstimation 41
2.4.3 BayesianPropertiesandEstimation 42
2.4.4 RComputing 43
2.4.5 HypothesisTestsforDispersion 50
vii
viii Contents
2.5 GeneratingData 52
2.5.1 InversionMethod 52
2.5.2 RejectionSampling 53
2.5.3 RComputing 56
2.6 ReparametrizedForms 57
2.7 COM–PoissonIsaWeightedPoissonDistribution 64
2.8 ApproximatingtheNormalizingTerm,Z(λ,ν) 65
2.9 Summary 69
3 DistributionalExtensionsandGeneralities 71
3.1 TheConway–Maxwell–Skellam(COM–SkellamorCMS)
Distribution 72
3.2 TheSum-of-COM–Poissons(sCMP)Distribution 75
3.3 Conway–MaxwellInspiredGeneralizations
oftheBinomialDistribution 77
3.3.1 TheConway–Maxwell–binomial(CMB)Distribution 78
3.3.2 The Generalized Conway–Maxwell–Binomial
Distribution 81
3.3.3 The Conway–Maxwell–multinomial (CMM)
Distribution 83
3.3.4 CMBandCMMasSumsofDependentBernoulli
RandomVariables 86
3.3.5 RComputing 87
3.4 CMP-MotivatedGeneralizationsoftheNegativeBinomial
Distribution 93
3.4.1 TheGeneralizedCOM–Poisson(GCMP)Distribution 93
3.4.2 TheCOM–NegativeBinomial(COMNB)Distribution 98
3.4.3 The COM-type Negative Binomial (COMtNB)
Distribution 101
3.4.4 ExtendedCMP(ECMP)Distribution 106
3.5 Conway–MaxwellKatz(COM–Katz)ClassofDistributions 112
3.6 FlexibleSeriesSystemLife-LengthDistributions 113
3.6.1 TheExponential-CMP(ExpCMP)Distribution 114
3.6.2 TheWeibull–CMP(WCMP)Distribution 117
3.7 CMP-Motivated Generalizations of the Negative
HypergeometricDistribution 119
3.7.1 The COM-negative Hypergeometric (COMNH)
Distribution,TypeI 120
3.7.2 TheCOM–Poisson-typeNegativeHypergeometric
(CMPtNH)Distribution 120