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

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104 Actuarial Modelling of Claim Counts regression coefficients presented in Chapter 1. The model-based estimation of ̂ in the GEE case is given by ̂ = −1 GEE where GEE = n∑ i=1 ( ) T ( ) EN −1 i V i EN i Here, −1 GEE is the GEE-equivalent of the inverse of the Fisher information matrix. It is a consistent estimator of the covariance matrix of ̂ if the mean structure and the ‘working’ correlation matrix are correctly specified. Then, ̂ = −1 GEE GEE −1 GEE is the robust estimate of the covariance matrix of ̂, where GEE = n∑ i=1 ( ) T ( ) EN i V −1 −1 i CN i V i EN i The robust estimate for ̂ is consistent even if the ‘working’ correlation matrix is misspecified. In computing, the covariance matrix CN i of N i is replaced with ĈN i = N i − ÊN iN i − ÊN i T The robust estimated variance-covariance matrix of the estimated regression coefficients is ̂̂ = ⎛ ⎜ ⎝ 0003910 −0000359 −0003418 −0003393 −0000509 −0000559 −0000359 0000801 0000138 0000081 −0000054 −0000015 −0003418 0000138 0003755 0003395 −0000089 −0000034 −0003393 0000081 0003395 0003806 −0000051 −0000029 −0000509 −0000054 −0000089 −0000051 0001100 0000595 −0000559 −0000015 −0000034 −0000029 0000595 0001132 If we compare Tables 2.10 (where the serial independence was assumed) and 2.12 (where the serial dependence is taken into account), we see that the standard errors are systematically larger in the GEE approach as serial dependence induces overdispersion. Generalized score tests for Type III contrasts are computed for GEE models (Wald tests are also available). Results of the Type 3 analysis are as follows: ⎞ ⎟ ⎠ Source DF Chi-square Pr>Chi-sq Gender 1 575 00165 Age ∗ Power 2 12578
Risk Classification 105 Table 2.12 Results of the Poisson regression with a GEE approach, Portfolio B. Variable Level Coeff Std error 95 % conf limit Z Pr > Z Intercept −12506 00625 −13731 −11280 −2000
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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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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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