transformation kernel estimation of insurance claim cost distributions

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1、TransformationkernelestimationofinsuranceclaimcostdistributionsCatalinaBolance,MontserratGuill´en,andJensPerchNielsen´Abstract.Atransformationkerneldensityestimatorthatissuitableforheavy-taileddistribu-tionsisdiscussed.Usingatruncatedbetatransformation,thechoiceofthebandwidthparameterbecomesst

2、raightforward.Anapplicationtoinsurancedataandthecalculationofthevalue-at-riskarepresented.Keywords:non-parametricstatistics,actuariallossmodels,extremevaluetheory1IntroductionTheseverityofclaimsismeasuredinmonetaryunitsandisusuallyreferredtoasinsurancelossorclaimcostamount.Theprobabilitydensit

3、yfunctionofclaimamountsisusuallyrightskewed,showingabigbulkofsmallclaimsandsomerelativelyinfrequentlargeclaims.Foraninsurancecompany,densitytailsarethereforeofspecialinterestduetotheireconomicmagnitudeandtheirinfluenceonre-insuranceagreements.Itiswidelyknownthatlargeclaimsarehighlyunpredictable

4、whiletheyarere-sponsibleforfinancialinstabilityandso,sincesolvencyisamajorconcernforbothinsurancemanagersandinsuranceregulators,thereisaneedtoestimatethedensityofclaimcostamountsandtoincludetheextremesinalltheanalyses.Thispaperisaboutestimatingthedensityfunctionnonparametricallywhendataareheavy

5、-tailed.Otherapproachesarebasedonextremes,asubjectthathasreceivedmuchattentionintheeconomicsliterature.Embrechtsetal.,Coles,andReissandThomas[8,11,15]havediscussedextremevaluetheory(EVT)ingeneral.Chavez-DemoulinandEmbrechts[6],basedonChavez-DemoulinandDavison[5],havediscussedsmoothextremalmode

6、lsininsurance.Theyfocusedonhighlightingnonparametrictrends,asatimedependenceispresentinmanycatastrophicrisksituations(suchasstormsornaturaldisasters)andinthefinan-cialmarkets.ArecentworkbyCoorayandAnanda[9]combinesthelognormalandtheParetodistributionandderivesadistributionwhichhasasuitableshape

7、forsmallclaimsandcanhandleheavytails.OthershaveaddressedthissubjectM.Corazzaetal.(eds.),MathematicalandStatisticalMethodsforActuarialSciencesandFinance©Springer-VerlagItalia201044C.Bolance,M.Guill´en,andJ.P.Nielsen´withtheg-and-hdistrib