Generalized linear models for insurance data

208 Pages·2013·4.599 MB·English

by  De Jong, Piet;Heller, Gillian Z

GeneralizedLinearModelsforInsuranceData
Actuaries should have the tools they need. Generalized linear models are
usedintheinsuranceindustrytosupportcriticaldecisions. Yetnotextintro-
ducesGLMsinthiscontextandaddressesproblemsspecifictoinsurancedata.
Untilnow.
Practicalandrigorous,thisbookstreatsGLMs,coversallstandardexponen-
tialfamilydistributions,extendsthemethodologytocorrelateddatastructures,
anddiscussesothertechniquesofinterestandhowtheycontrastwithGLMs.
Thefocusisonissueswhicharespecifictoinsurancedataandalltechniques
areillustratedondatasetsrelevanttoinsurance.
Exercises anddata-basedpracticals helpreaderstoconsolidatetheirskills,
with solutions and data sets given on the companion website. Although the
book is package-independent, SAS code and output examples feature in an
appendixandonthewebsite. Inaddition,Rcodeandoutputforallexamples
areprovidedonthewebsite.
InternationalSeriesonActuarialScience
MarkDavis,ImperialCollegeLondon
JohnHylands,StandardLife
JohnMcCutcheon,Heriot-WattUniversity
RagnarNorberg,LondonSchoolofEconomics
H.Panjer,WaterlooUniversity
AndrewWilson,WatsonWyatt
The International Series on Actuarial Science, published by Cambridge
UniversityPressinconjunctionwiththeInstituteofActuariesandtheFaculty
ofActuaries,willcontaintextbooksforstudentstakingcoursesinorrelatedto
actuarialscience,aswellasmoreadvancedworksdesignedforcontinuingpro-
fessionaldevelopmentorfordescribingandsynthesizingresearch. Theseries
willbeavehicleforpublishingbooksthatreflectchangesanddevelopmentsin
thecurriculum,thatencouragetheintroductionofcoursesonactuarialscience
in universities, and that show how actuarial science can be used in all areas
wherethereislong-termfinancialrisk.

GENERALIZED LINEAR
MODELS FOR
INSURANCE DATA
PIET DE JONG
DepartmentofActuarialStudies,MacquarieUniversity,Sydney
GILLIAN Z. HELLER
DepartmentofStatistics,MacquarieUniversity,Sydney
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,Sa˜oPaulo,Delhi
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:
http://www.afas.mq.edu.au/research/books/glms for insurance data
(cid:2)c P.deJongandG.Z.Heller2008
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2008
Thirdprinting2009
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
ISBN978-0-521-87914-9hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication,
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.Informationregardingprices,traveltimetablesandother
factualinformationgiveninthisworkarecorrectatthetimeoffirstprintingbut
CambridgeUniversityPressdoesnotguaranteetheaccuracyofsuch
informationthereafter.
Contents
Preface pageix
1 Insurancedata 1
1.1 Introduction 2
1.2 Typesofvariables 3
1.3 Datatransformations 4
1.4 Dataexploration 6
1.5 Groupingandrunofftriangles 10
1.6 Assessingdistributions 12
1.7 Dataissuesandbiases 13
1.8 Datasetsused 14
1.9 Outlineofrestofbook 19
2 Responsedistributions 20
2.1 Discreteandcontinuousrandomvariables 20
2.2 Bernoulli 21
2.3 Binomial 22
2.4 Poisson 23
2.5 Negativebinomial 24
2.6 Normal 26
2.7 Chi-squareandgamma 27
2.8 InverseGaussian 29
2.9 Overdispersion 30
Exercises 33
3 Exponentialfamilyresponsesandestimation 35
3.1 Exponentialfamily 35
3.2 Thevariancefunction 36
3.3 Proofofthemeanandvarianceexpressions 37
3.4 Standarddistributionsintheexponentialfamilyform 37
3.5 Fittingprobabilityfunctionstodata 39
Exercises 41
v
vi Contents
4 Linearmodeling 42
4.1 Historyandterminologyoflinearmodeling 42
4.2 Whatdoes“linear”inlinearmodelmean? 43
4.3 Simplelinearmodeling 43
4.4 Multiplelinearmodeling 44
4.5 Theclassicallinearmodel 46
4.6 Leastsquarespropertiesundertheclassicallinearmodel 47
4.7 Weightedleastsquares 47
4.8 Groupedandungroupeddata 48
4.9 Transformationstonormalityandlinearity 49
4.10 Categoricalexplanatoryvariables 51
4.11 Polynomialregression 53
4.12 Bandingcontinuousexplanatoryvariables 54
4.13 Interaction 55
4.14 Collinearity 55
4.15 Hypothesistesting 56
4.16 Checksusingtheresiduals 58
4.17 Checkingexplanatoryvariablespecifications 60
4.18 Outliers 61
4.19 Modelselection 62
5 Generalizedlinearmodels 64
5.1 Thegeneralizedlinearmodel 64
5.2 Stepsingeneralizedlinearmodeling 65
5.3 Linksandcanonicallinks 66
5.4 Offsets 66
5.5 Maximumlikelihoodestimation 67
5.6 Confidenceintervalsandprediction 70
5.7 Assessingfitsandthedeviance 71
5.8 Testingthesignificanceofexplanatoryvariables 74
5.9 Residuals 77
5.10 Furtherdiagnostictools 79
5.11 Modelselection 80
Exercises 80
6 Modelsforcountdata 81
6.1 Poissonregression 81
6.2 Poissonoverdispersionandnegativebinomialregression 89
6.3 Quasi-likelihood 94
6.4 Countsandfrequencies 96
Exercises 96
7 Categoricalresponses 97
7.1 Binaryresponses 97
7.2 Logisticregression 98
Contents vii
7.3 Applicationoflogisticregressiontovehicleinsurance 99
7.4 Correctingforexposure 102
7.5 Groupedbinarydata 105
7.6 Goodnessoffitforlogisticregression 107
7.7 Categoricalresponseswithmorethantwocategories 110
7.8 Ordinalresponses 111
7.9 Nominalresponses 116
Exercises 119
8 Continuousresponses 120
8.1 Gammaregression 120
8.2 InverseGaussianregression 125
8.3 Tweedieregression 127
Exercises 128
9 Correlateddata 129
9.1 Randomeffects 131
9.2 Specificationofwithin-clustercorrelation 136
9.3 Generalizedestimatingequations 137
Exercise 140
10 Extensionstothegeneralizedlinearmodel 141
10.1 Generalizedadditivemodels 141
10.2 Doublegeneralizedlinearmodels 143
10.3 Generalizedadditivemodelsforlocation,scaleandshape 143
10.4 Zero-adjustedinverseGaussianregression 145
10.5 Ameananddispersionmodelfortotalclaimsize 148
Exercises 149
Appendix1 Computercodeandoutput 150
A1.1 Poissonregression 150
A1.2 Negativebinomialregression 156
A1.3 Quasi-likelihoodregression 159
A1.4 Logisticregression 160
A1.5 Ordinalregression 169
A1.6 Nominalregression 175
A1.7 Gammaregression 178
A1.8 InverseGaussianregression 181
A1.9 LogisticregressionGLMM 183
A1.10 LogisticregressionGEE 185
A1.11 LogisticregressionGAM 187
A1.12 GAMLSS 189
A1.13 Zero-adjustedinverseGaussianregression 190
Bibliography 192
Index 195