A HIERARCHICAL POISSON MODEL FORCLAIM FREQUENCY DATAHing Wu Jing ZhaoDecember 8, 20071 IntroductionThis project is about two ways to predict future claim count in a hierarchicalmo del The first method is the non- parametric Buhlmann-Straub credibilitymo del The second method is Bayesian Graphical Modelling method. We triedto compare the two methods by looking at the statistics such as mean, standarderror and credible set.2 Credibility TheoryIn actuarial field the term credibility was originally attached to experience ratingformulas that were convex combinations (weighted averages) of individual andclass estimates of the individual risk premium. Credibility theory, thus, wasthe branch of actuarial mathematics that discovered model-based principles forconstruction of such formulas. The risk premium m, defined as the expectedclaims expenses per unit of risk exposed, for an individual risk selected from ap ortfolio (class) of similar risks. Advocating the combined use of individual riskexperience and class risk experience. The premium rate is a weighted averageof the formM = m ∗ z + (1 − z) ∗ u (1)where m is the observed mean claim amount per unit of risk exposed for theindividual contract and u is the corresponding overall mean in the insurancep ortfolio. In the language of modern credibility theory, it is a function m(Θ) ofa random element Θ representing the unobservable characteristics of the indi-vidual risk. The random nature of Θ expresses the notion of heterogeneity; theindividual risk is a random selection from a portfolio of similar but not iden-tical risks, and the distribution of Θ describes the variation of individual riskcharacteristics across the portfolio. The weight z was soon to be named credi-bility factor since it measures the amount of credence attached to the individualexperience, and M was called the credibility premium.12.1 B¨uhlmann-Straub credibility modelIn this article, suppose that an insurance company has seven groups of policy-holder’s loss frequency counts for 50 months as shown in Table 1. The problemis to describe the loss frequencies anticipated in the 51th month. One approachis to make a straightforward application of the Buhlmann-Straub credibilitymo del as described in Herzog (1996, chap. 7) and Klugman, Panjer, and Will-mot (1998, sections 5.4.4 and 5.5.1). The non-parametric method described inthese texts can be used to estimate the necessary population parameters andcredibility factors. In terms of the notation found in Herzog (1996) these calcu-lation of values like m, a, v, z, m i, u yield the predicted count frequence of the51th month.u = E(E(Xj|Θ)) (2)v = E(V ar(Xj|Θ)) (3)a = V an(E(Xj|Θ)) (4)z = mi/(mi− v/a) (5)Table 1:Policy Group Month1 Month2 .......... Month 51Claim Counts 1 0 4 .......... ?No. in group 236 191 .......... 168Claim Counts 2 28 7 .......... ?No. in group 252 186 .......... 184Claim Counts 3 1 6 .......... ?No. in group 208 184 .......... 191Claim Counts 4 23 2 .......... ?No. in group 243 187 .......... 262Claim Counts 5 0 1 .......... ?No. in group 178 202 .......... 186Claim Counts 6 0 6 .......... ?No. in group 160 181 .......... 222Claim Counts 7 18 4 .......... ?No. in group 207 155 .......... 20223 Bayesian Graphical Modeling by Markov ChainMonte Carlo SimulationsAnother approach would be to treat the unknown parameters as random vari-ables and then develop the posterior and predictive distributions of interestusing the Bayesian method.consider a vector random variable U (U1, . . . , Uk) with joint distributionf(U1, . . . , Uk). In a Bayesian context, some of these variables are modelparameters while others may represent unobserved past or future data. Supposef(U) has a complicated and analytically intractable form, and the expected valueof some integrable function h(U) is sought. Even if this calculation can not bep erformed analytically, it is still possible that the probabilistic model asso ciatedwith f(U) may be simple enough to permit independent random draws U(t), t1, . . . , n, from it. If this is the case, then the desired expectation can beapproximated usingE[h(u)] ≈1nnXi=1h(ut)For instance, the population mean can be estimated using the sample mean.This procedure is called Monte Carlo integration. Unfortunately, many compli-cated models will not readily permit independent random draws. In this casea MCMC simulation method can be used instead. The main idea behind aMCMC method is to simulate realizations from a Markov chain that has f(U)as its stationary distribution.we adopt the following complete probability model for the data appearingin Table 1. Let Xijand Pijdenote the number of claims and number of people,respectively, for the i-th group policyholder in the j-th policy month. We assumethat all Xijare conditionally independent and that Xij∼ P oisson(Pijθi), forall i and j. The parameters λidenote the exp ected number of losses per unitof exposure for group policyholder i. Given α and β, the θiare assumed tob e conditionally independent with θi∼ gamma(α, β), for all i. These gammadistributions are parameterized to have mean α/β. We complete this model byletting α ∼ gamma(5, 5) and β ∼ gamma(25, 1). These last two distributionswere selected arbitrarily. However, they imply that each ui has a prior mean andstandard deviation approximately equal to 0.041 and 0.048 respectively, whichis not unreasonable for the present context.We start with a consideration of the following slight extension of the fullprobability model:Xij∼ P oisson (λij) ,λij= Pijθij,θi∼ gamma (α, β) ,α ∼ gamma (5, 5) ,β ∼ gamma (25, 1) ,3Pij∼ P oisson (ai) ,ai∼ Unif orm (100, 300) ,For the data in Table 1, the values taken on by the indices are i = 1, 2, ..., 7and j = 1, ..., 51 This is the form of the model we will implement using BUGS/WinBUGS.We assume conditional independence for the variables appearing on the left-hand side of each line given the model parameters appearing on the right. Sothe variables Xijare conditionally independent given the values of λij. The firstfive lines describe the original hierarchical Poisson model. The other three linesdescribe a simple probability model for the number of people.The model has the particularly simple graphical representation given in Fig-ure 1. This graph emphasizes the main qualitative features of the model withoutthe use of cumbersome algebraic formulas. Nodes in the graph denote the dataand parameters appearing in the model.
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