非寿险精算Loss-number-distributionPPT课件.pptx

非寿险精算非寿险精算Loss number distributionQuestions Suppose the numbers of payment for some medical insurance policy follows Poisson distribution,now modified it with deductible of￥100，which distribution can be used to describe the numbers of payment for the modified policy?And which distribution for the case that the insurer issued a reinsurance program for this policy?Ch2.1 The(a,b,0)ClassLet N be a random variable representing the number of some events.Denote its probability function asThe(a,b,0)class:If there exists constants a and b such thatThen N is a member of the(a,b,0)class.Include and only include Poisson distributionNegative Binomial distributionBinomial distributionCh2.1.1 Poisson DistributionNPoisson()EN=VarN=Pgf AdditivityDecomposability Ch2.1.2 Negative Binomial Dis.EN=VarN=Pgf Generalization of Poisson distributionMixingCompound Limiting leads to Poisson DistributionCh2.1.2 Geometric DistributionSpecial case of negative binomial dis.when r=1,i.e.Memoryless property Given that there are at least m claims,the probability distribution of the number of claims in excess of m does not depend on m.Ch2.1.3 Binomial DistributionEN=VarN=PgfRandom variable with binomial distribution has the finite maximum value,so it can be used for the case with finite support.Ch2.1 The(a,b,0)ClassCh2.2 The(a,b,1)classThe(a,b,1)class:If there exists constants a and b such thatThen N is a member of the(a,b,1)class.zero-truncatedzero-modifiedan(a,b,0)dis.+a degenerated dis.A zero-truncated(a,b,1)dis.+a degenerated dis.ETNB distributionCh2.3 Counting Processthe number of payments in time intervalsrespectively For any t in 0,T,the number of payments isA stochastic process is a sequence of r.v.s or a sequence of r.v.s is increment of time interval s,tStationary increments:depends on t and s only through the difference t-s.Independent increments:increment of any set of disjoint intervals are independent.Ch2.3 Counting ProcessCh2.3.1 Birth ProcessBirth processNotes Ch2.3.2 Poisson ProcessLet transition intensity The process is homogeneous Poisson process Transition ProbabilityDis.of incrementDis.of each timenonhomogeneous Poisson processCh2.3.2 Poisson ProcessLet denote the time of the first event and the time between(n-1)st and the nth event is the sequence of inter-arrival timeCh2.3.3 Process with contagionLet transition intensity The process is process with contagion with does not depend on t:homogeneousfor n=0,1,from Theorem 6.9,we have Ch2.4 compound frequency modelLet be i.i.d random variables with pgf ,be a counting random variable with pfg The total claim is random sumThen S is called compound model Its pgf is Example:1 Zero-Modified dis.is a compound model 2 Poisson-Logarithmic compound model follows negative binomial distributionCh2.4 compound frequency modelDenoteTheorem 6.12 If the primary distribution is a member of the(a,b,0)class,the recursive formula is Theorem 6.13 If the primary distribution is a member of the(a,b,1)class,the recursive formula is Ch2.4 compound frequency modelTheorem 6.14Compound distribution is not always create new distribution.Theorem 6.15 Suppose the pgf satisfies then the pgf of compound process is Ch2.4.1 Compound Poisson modelPgfMeanVarianceSkewnessGiven the secondary pare the skewness Noncompound ENTB Negative BinomialPolya-Aeppli Neyman Type A Binomial Ch2.4.1 Compound Poisson model Theorem 6.16 Suppose that Si has compound Poisson dis.with parameter and secondary dis.as and Si are independent with each other.Then has compound Poisson dis.with parameter and secondary dis.asExample:Determine the distribution of sum ofnegative binomial distribution r.v.s Ch2.5.1 Mixed ModelPgfPdf is called risk parameter,and its dis.is called risk parameter dis./mixing dis.Connection with finite mixture distributionZero-modified distribution is a mixture dis.Beta dis.mixing with binomial dis.is negative hypergeometric dis.Ch2.5.2 Mixed PoissonPgfMean Variance difference mixing dis.can not lead to the same mixed Poisson distributionInfinitely divisible dis.Normal dis.,Gamma dis.,Poisson dis.,Negative binomial dis.If N follow mixed Poisson dis.,that N also follows compound Poisson dis.