Application Of Zero-inflated Poisson Distribution In Insurance Claiming Credibility Model

Application of Zero-inflated Poisson Distribution in Insurance ClaimingCredibility ModelIn the practical theory of insurance, pricing is very important, because it affects competitiveness of insurance company. In the process of pricing, the pure premium is the basic and hardest part. The pure premium P equals the claim frequency multiplies the claim amount. But the factors to influence these two elements are different, so we should establish two different forecasting models respectively. Now we expound the claim frequency, which is predicted by the Buhlmann-Straub model.In the process of settling historical data, we find the insurance claim data meet the discrete "zero" more. We will fit the claim number with zero-inflated Poisson, which improves frequently-used Poisson distribution. We draw zero-inflated Poisson into the Buhlmann-Straub model, and predict the future claim frequency.The subject is divided into four parts. In the second chapter, we introduce the basic knowledge of insurance pricing. Insurance pricing is the insurance rate set by actuarial data, using the actuarial method. The insurance premium can meet the target of long-term interest rates. It is comprised of pure premium and additional premium. Pure premium is the main and the hardest part. The additional premium can be expressed by the proportion of pure premium.Pricing the premium need some basic principles:The premium should meet sufficient principle,2. The premium should meet fairness principle,3. The premium should meet stability principle,4. The premium should meet compliance principle.The third chapter introduces zero-inflated Poisson distribution.Its mathematical expectation and variance are Using the method of moment estimating, the torque ofλand q areλand q. Eventually we get the fitting function for the zero-inflated distribution,The fourth chapter introduces the forecast model of claim frequency, the improved Buhlmann-Straub model.For the risk i, we can get the following information:Nij number of claims in year j,ij number of years at risk in year j,Fij= Nij/ξij claim frequency in year j.Buhlmann-Straub Model hypothesizes 4.4.1Condition 1:Conditionally, givenθi, Nij (j= 1,2……) are independently with zero-inflated distribution withλij(θi)=ξijθiλ0 and q.Condition 2:(θ1, N1), (θ2,N2)…are independently, andθ1,θ2…are independent and identically distributed.Theorem 4.2.1 (nonhomogeneous credibility estimate).Under the assumption hypothesizes 4.4.1, Buhlmann-Straub Model is given by Theorem 4.2.2 (homogeneous credibility estimate).Under the assumption hypothesizes 4.4.1, the homogeneous credibility estimate of claim frequency is given byIn process of on-life insurance pricing, the credibility model meet the unbiased for total claim amout in the same risk.Theorem 4.2.5 (balance property).Under the assumption hypothesizes 4.4.1, we have for the homogeneous credibility estimate of claim frequency in the Buhlmann-Straub modelThe homogeneous credibility estimate of claim frequency in the Buhlmann-Straub model have three unknown parametersσ2,τ2 andγ2. We estimate them forAccording to the theory of Buhlmann-Straub model, we use the claim data of a property company from March to December in 2004 to analysis the improved model.When the claim number obey the zero-inflated Poisson distribution,Finally, we predict the claim frequency of the comprehensive risk and DLW, (2) When the claim number obey the Poisson distribution,Finally, we predict the claim frequency of the comprehensive risk and DLW,Obviously, when the claim number obey the Poisson distribution, the claim frequency is dependent on Fi(?). The reason is the risk doesn't remove the claim number of zero which makesξi(?) is distinctly larger than (?). So the credibility factor (?)closes to 1.