The Application Of Zero-Inflated Poisson Distribution In Insurance

I .Zero-inflated Poisson Distribution(a) .Poisson regression models are commonly used to fit the count data in several real-life examples . But in some applications concerning discrete data ,the number of zeros is observed to be higher than that can be modeled with the usual Poisson probability model. Because the no claims discount (NCD) systems are common in the automobile insurance market,the policy-holder have to decide whether to purchase the insurance offered by the firm . When we modelling the counts of indemnity collected from insurance company,there were usually a relatively large number of zeros. But the number of the counts will be large in case that the insured have purchased the insurance .To deal with such data,we considered a discrete random variable Y with a zero-inflated Poisson distribution in this paper .It assumes that with probability p the only possible observation is 0,and with probability 1—p a Poisson(λ)random variable is observed.So that(b) .If the generalized Poisson distribution is mixed with zero,we have a new distributionWe call it zero-inflated generalized Poisson distribution,which include three parameters p,λ,α. We also know that p∈((1 - e^λ)^-1,1),α∈[0, 1/λ)andλ> 0.Note that if p > 0,α= 0,then Y - ZIP(λ,p).Maximum likelihood method and Bayesian method are used to estimate the parameters respectively.(c) .In this paper ,we discussed ZIP model with covariates .Both the probability p of the zero state and the meanλof the Poisson state depend on covariates.The responses y = (y₁, y₂,... ,y_n)' are independent and Moreover,the parameters satisfyWe show how to compute the MLE's whenλand p are functionally related.We also gave the MLE's when they are not related.(c) .A simulation with 200,000 data was given in this paperWe have the tableZIP model and ZIGP model have been shown to be useful for modeling this situation where count data with many zeros are encountered.And we only introduced ZIP distribution into the insurance models.II .The applications of ZIP Distribution in Insurance models(a) . First of all,we introduced the definition of compound zero-inflated Poisson distribution.Definition 2.1.1 The random variable C_i has a compound zero-inflated Poisson distribution with parameters p,λ.if1. N,C₁,C₂,... are independent,2. C₁,C₂,... have the same distribution,3. N has a ZIP distribution with parameters p,λ. Theorem 2.1.1 If the random variable C_i has a compound zero-inflated Poisson distribution with parameters p andλ,then we have(b) . In this paper we also discussed the asymptotic distribution of collective claimsin in Automobile Insurance.Theorem 2.2 1 If the random variable C_i has a compound zero-inflated Poisson distribution with parameters p,λ,andthen the distribution of Z approach to standard asymptotic normal distribution whenλâ†'∞,pâ†'0 andλpâ†'0.Sometimes,the compound zero-inflated Poisson distribution is asymmetrical.We have to choose a distribution which is not symmetric to approach the distribution of 5.Theorem 2.2.2 If the random variable C_i has a compound zero-inflated Poisson distribution with parameters p,λ,and p₃-pλ²p₁³+ 2p²Î»²p₁³ + 3pp₁p₂λ> 0.thensinceWe discussed these two asymptotic models in the case that p = 0.1,λ= 16 and p = 0.3,λ= 5. (c) . The paper considered the credibility theory in non-life insurance ,and discussed the expression of the credibility ZIn the case of asymptotic normal distribution ,we haveWe often used the NP asymptotic method and letsince . The credibility Z can be obtained byWe also gave an example that random variable N - ZIP(p = 0.6,λ= 0.9),and we figured out Z when q's, k's were different.