Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

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110 Actuarial Modelling of Claim Counts • when we compare the Poisson-Gamma model with the Poisson-Inverse Gaussian model, the value of the test statistic is equal to −07988 leading to a p-value of 4244 %. • when we compare the Poisson-Gamma model with the Poisson-LogNormal model, the value of the test statistic is equal to −07688 leading to a p-value of 4420 %. • when we compare the Poisson-Inverse Gaussian model with the Poisson-LogNormal model, the value of the test statistic is equal to −05922 leading to a p-value of 5537 %. Therefore, the three models are not statistically different. Applying the same testing procedure to the policies that are in the portfolio for the first two years (this means 12 202 policies) yields the same conclusion. 2.9.10 Information Criteria Non-nested models are often compared using likelihood-based criteria, including the well-known AIC, for instance. Since the competing models have the same number of parameters, it is enough to examine the respective log-likelihoods. A commonly used rule-of-thumb consists in considering that two models are significantly different if the difference in the log-likelihoods exceeds five (corresponding to a difference in AICs of more than ten, as discussed in Burnham & Anderson (2002)). This means here that the Poisson-Inverse Gaussian and Poisson-LogNormal models are significantly better than the Negative Binomial model. Considering the maximum of the log-likelihood, we chose the Poisson-LogNormal model for Portfolio B. The same conclusion is obtained using different criteria often used in practice. For instance, Raftery (1995) suggested that a model significantly outperfoms a competitor if the difference in their respective BIC values exceeds 5. 2.9.11 Resulting Classification for Portfolio B Table 2.16 gives the resulting price list obtained with the Poisson-LogNormal model described in Table 2.15. The final a priori ratemaking contains 18 classes. This ratemaking will be used throughout the text for the examples involving Portfolio B. There is another way to present the results displayed in Table 2.16. The annual expected claim frequency is obtained from the reference class, with exp̂ 0 = 2950 % according to Table 2.15. Applying correction coefficients, we then obtain the annual expected claim frequency of any policyholder. Specifically, it is obtained from the multiplicative formula
Risk Classification 111 Table 2.16 A priori risk classification for Portfolio B (Poisson-LogNormal regression model). Age–Power Size of the city Gender Annual claim freq. (%) Group 1 Group 2 Group 3 Large Middle Small Female Male Weights (%) No No Yes No No Yes No Yes 2950 082 No No Yes No No Yes Yes No 2759 043 No No Yes No Yes No No Yes 3173 063 No No Yes No Yes No Yes No 2968 037 No No Yes Yes No No No Yes 3833 044 No No Yes Yes No No Yes No 3586 031 No Yes No No No Yes No Yes 1960 788 No Yes No No No Yes Yes No 1833 464 No Yes No No Yes No No Yes 2108 832 No Yes No No Yes No Yes No 1972 466 No Yes No Yes No No No Yes 2547 689 No Yes No Yes No No Yes No 2382 399 Yes No No No No Yes No Yes 1519 1259 Yes No No No No Yes Yes No 1420 702 Yes No No No Yes No No Yes 1634 1381 Yes No No No Yes No Yes No 1528 724 Yes No No Yes No No No Yes 1974 1268 Yes No No Yes No No Yes No 1846 730 2950 % { exp−00669 = 094 if the policyholder is a female driver × 1 otherwise ⎧ exp−06640 = 051 if the policyholder belongs to Group 1 with respect to ⎪⎨ the combined variable Age ∗ Power × exp−04089 = 066 if the policyholder belongs to Group 2 with respect to the combined variable Age ⎪⎩ ∗ Power 1 otherwise ⎧ ⎨ exp02620 = 130 if the policyholder resides in a large city × exp00731 = 108 if the policyholder resides in a middle-sized city ⎩ 1 otherwise 2.10 Further Reading and Bibliographic Notes 2.10.1 Generalized Linear Models After decades dominated by statistically unsophisticated models, it is now common practice since Mc Cullagh & Nelder (1989) and Brockman & Wright (1992) to achieve a priori classification with the help of generalized linear models (GLMs). They are so called because they generalize the classical linear models based on the Normal distribution.
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Contents Foreword Preface Notation
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Contents vii 2.4 Overdispersion 79
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Contents ix 4.2.3 Trajectory 170 4.
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Contents xi 7.4 Numerical Illustrat
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Part I Modelling Claim Counts Actua
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Credibility Models for Claim Counts
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Part III Advances in Experience Rat
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8 Transient Maximum Accuracy Criter
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Transient Maximum Accuracy Criterio
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Bibliography Albrecht, P. (1983a).
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Bibliography 347 Daengdej, J., Luko
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Bibliography 349 Greenwood, M., & Y
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Bibliography 351 Norberg, R. (1976)
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Bibliography 353 Walhin, J.-F., & P
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