非寿险精算Loss-number-distribution课件

非寿险精算LossnumberdistributionQuestionsSupposethenumbersofpaymentforsomemedicalinsurancepolicyfollowsPoissondistribution,nowmodifieditwithdeductibleof￥100，whichdistributioncanbeusedtodescribethenumbersofpaymentforthemodifiedpolicy?Andwhichdistributionforthecasethattheinsurerissuedareinsuranceprogramforthispolicy?Ch2.1The(a,b,0)ClassLetNbearandomvariablerepresentingthenumberofsomeevents.DenoteitsprobabilityfunctionasThe(a,b,0)class:Ifthereexistsconstantsa

andbsuchthatThenNisamemberofthe(a,b,0)class.Includeandonlyinclude

PoissondistributionNegativeBinomialdistributionBinomialdistributionCh2.1.1PoissonDistributionN~Poisson()E[N]=Var[N]=PgfAdditivityDecomposability

Ch2.1.2NegativeBinomialDis.E[N]=＜Var[N]=PgfGeneralizationofPoissondistributionMixingCompoundLimitingleadstoPoissonDistributionCh2.1.2GeometricDistributionSpecialcaseofnegativebinomialdis.whenr=1,i.e.MemorylesspropertyGiventhatthereareatleastmclaims,theprobabilitydistributionofthenumberofclaimsinexcessofmdoesnotdependonm.Ch2.1.3BinomialDistributionE[N]=＞Var[N]=PgfRandomvariablewithbinomialdistributionhasthefinitemaximumvalue,soitcanbeusedforthecasewithfinitesupport.Ch2.1The(a,b,0)Class

Ch2.2The(a,b,1)classThe(a,b,1)class:IfthereexistsconstantsaandbsuchthatThenNisamemberofthe(a,b,1)class.zero-truncatedzero-modifiedan(a,b,0)dis.+adegenerateddis.Azero-truncated(a,b,1)dis.+adegenerateddis.ETNBdistribution

Ch2.3CountingProcessthenumberofpaymentsintimeintervals

respectivelyForanytin[0,T],thenumberofpaymentsisAstochasticprocessisasequenceofr.v.sorasequenceofr.v.sisincrementoftimeinterval[s,t]Stationaryincrements:dependsontandsonlythroughthedifferencet-s.Independentincrements:incrementofanysetofdisjointintervalsareindependent.Ch2.3CountingProcess

Ch2.3.1BirthProcessBirthprocessNotesCh2.3.2PoissonProcessLettransitionintensityTheprocessishomogeneousPoissonprocess

TransitionProbabilityDis.ofincrementDis.ofeachtimenonhomogeneousPoissonprocessCh2.3.2PoissonProcessLetdenotethetimeofthefirsteventandthetimebetween(n-1)standthentheventisthesequenceofinter-arrivaltimeCh2.3.3ProcesswithcontagionLettransitionintensityTheprocessisprocess

withcontagion

withdoesnotdependont:homogeneousforn=0,1,…,fromTheorem6.9,wehaveCh2.4compoundfrequencymodelLetbei.i.drandomvariableswithpgf,beacountingrandomvariablewithpfgThetotalclaimisrandomsumThenSiscalledcompoundmodel

ItspgfisExample:1

Zero-Modifieddis.isacompoundmodel2Poisson-LogarithmiccompoundmodelfollowsnegativebinomialdistributionCh2.4compoundfrequencymodelDenoteTheorem6.12Iftheprimarydistributionisamemberofthe(a,b,0)class,therecursiveformulaisTheorem6.13Iftheprimarydistributionisamemberofthe(a,b,1)class,therecursiveformulais

Ch2.4compoundfrequencymodelTheorem6.14Compounddistributionisnotalwayscreatenewdistribution.Theorem6.15Supposethepgfsatisfies

thenthepgfofcompoundprocessisCh2.4.1CompoundPoissonmodelPgfMeanVarianceSkewnessGiventhesecondarydis.comparetheskewnessNoncompoundENTBNegativeBinomialPolya-AeppliNeymanTypeABinomialCh2.4.1CompoundPoissonmodel

Theorem6.16SupposethatSihascompoundPoissondis.withparameterandsecondarydis.asandSiareindependentwitheachother.ThenhascompoundPoissondis.withparameterandsecondarydis.asExample:Determinethedistributionofsumof

negativebinomialdistributionr.v.sCh2.5.1MixedModelPgfPdfiscalledriskparameter,anditsdis.iscalledriskparameterdis./mixingdis.ConnectionwithfinitemixturedistributionZero-modifieddistributionisamixturedis.Betadis.mixingwithbinomialdis.isnegativehypergeometricdis.Ch2.5.2MixedPoissonPgfMeanVaria