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

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