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

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74 Actuarial Modelling of Claim Counts Table 2.1 Results of the Poisson regression for the model with the 5 explanatory variables, Portfolio A. Variable Level Coeff Std error Wald 95 % conf limit Chi-sq Pr>Chi-sq Intercept −22131 00582 −23271 −20991 144740 60 years −00010 02350 −04616 04596 000 09967 Gender ∗ Age Male 18–24 years 06429 00797 04867 07990 6510
Risk Classification 75 used). The ratio of the deviance to the number of degrees of freedom should be close to 1 to indicate goodness-of-fit. Here, we obtain 05357 (the deviance is equal to 7764.83). Note however that the data should be grouped to make the Chi-square approximation more reliable, so that we cannot use this statistic at the present stage. An important aspect of insurance ratemaking with generalized regression models is the selection of explanatory variables in the model. Changes in goodness-of-fit statistics are often used to evaluate the contribution of subsets of explanatory variables to a particular model. One strategy for variable selection is to fit a sequence of models, beginning with a simple model with only an intercept term, and then include one additional explanatory variable in each successive model. The importance of the additional explanatory variable can be measured by the difference in fitted log-likelihoods between successive models. Asymptotic tests computed by the GENMOD procedure enable the actuary to assess the statistical significance of the additional term (this is called Type I analysis in SAS R ). Another strategy (adopted here) consists of starting from a model incorporating all the available information, and then excluding the irrelevant explanatory variables. To this end, the GENMOD procedure generates a Type 3 analysis (analogous to Type III sums of squares in the GLM procedure). A Type 3 analysis does not depend on the order in which the terms for the model are specified (in contrast to the Type 1 analysis). Type 3 analysis compares the complete model (that is the model which includes all the specified variables) with the different submodels obtained by deleting one of the explanatory variables. It enables the actuary to test the relevance of one variable taking all the others into account. It roughly corresponds to the backward approach: at each step, we exclude the variable with the largest p-value until no more can be excluded (i.e. until all the p-values are smaller than a fixed threshold, generally 5 %). Note that the Type 3 analysis works with the variables and not with the levels of these variables. Indeed, it is possible to obtain a relevant variable for which some levels are not relevant. The results of the Type 3 analysis are as follows: Source DF Chi-square Pr > Chi-sq Gender ∗ Age 7 7516
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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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