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

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76 Actuarial Modelling of Claim Counts The option ‘Estimate’ of GENMOD (which is similar to the option ‘Contrast’) can be used to assess the relevance of grouping the levels of the interaction Age–Gender 2 by 2. This option computes likelihood ratio statistics for user-defined contrasts, that is, linear functions of the parameters, and p-values based on their asymptotic Chi-square distributions. Here, the option ‘Estimate’ is used to test for the equality of the regression coefficients (that is, for H 0 : j1 = j2 against H 1 : j1 ≠ j2 for all the combinations j 1 j 2 corresponding to binary variables coding Age ∗ Gender). The test statistic is based on the ratio of the likelihoods corresponding to the models under H 0 and under H 1 . This grouping process can be summarized as follows: Step 1 the reference level ‘Male 31–60’ is first merged with ‘Female >60’ (p-value of 9967 %); Step 2 then ‘Male 25–30’ and ‘Female 18–24’ are grouped together (p-value of 8585 %); Step 3 ‘Male 31–60’, ‘Male > 60’ and ‘Female > 60’ are grouped together (p-value of 6613 %); Step 4 ‘Male 31–60’, ‘Male > 60’, ‘Female 31–60’ and ‘Female > 60’ are grouped together (p-value of 3517 %); Step 5 finally ‘Male 25–30’, ‘Female 18–24’ and ‘Female 25–30’ are grouped together (p-value of 1239 %). The level ‘Male 18–24’ cannot be grouped with other levels. After grouping, the variable Age ∗ Gender resulting from the interaction of Age with Gender has three levels: ‘Males 18–24’, ‘Females 18–30 and Males 25–30’, and ‘Males and Females over 30’. As expected, it accounts for the extra risk of young male drivers (aged between 18 and 24), and of the young drivers, whereas the reference level 3 is assigned to drivers over 30. The fit of the final model is shown in Table 2.2. The log-likelihood is now equal to −54842 and the Type 3 analysis gives the following results: Source DF Chi-square Pr>Chi-sq Gender ∗ Age 2 7166
Risk Classification 77 Table 2.2 Results of the Poisson regression for the final model, Portfolio A. Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq Intercept −21975 00466 −22888 −21062 222551
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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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Part III Advances in Experience Rat
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8 Transient Maximum Accuracy Criter
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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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