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

Risk Classification 85
and then write the empirical analogue of the last relation
n∑
i=1
( (ki
− d i expscore i ) 2
− ki − ( d i expscore i ) 2
)
= 0
Therefore, the estimator of is given by
1
â =̂ = ∑ n
i=1
( (ki
− d i expŝcore i ) 2
− ki
)
∑ n
i=1
(
di expŝcore i ) 2
where ŝcore i =˜x T i ̂, ̂ being the Poisson maximum likelihood estimator for . The estimators
̂ and ̂ are consistent in the Poisson mixture model, and are thus good starting values for
finding the maximum likelihood estimators.
2.5.2 Numerical Illustration
The Negative Binomial regression with categorical variables can be performed with the
SAS R /STAT procedure GENMOD which corrects the estimations for overdispersion. The
final model for Portfolio A is shown in Table 2.4. The interpretation of the different columns
is the same as in Section 2.3.15. Compared with the Poisson fit, we see that the estimated
j s are very similar, but the standard errors are larger in the Negative Binomial case.
The estimation of the parameter a by the method of moments is given by â = 12401
whereas the maximum likelihood estimator is equal to â = 1065. The log-likelihood is equal
to −54485. The variance-covariance matrix of the estimated regression coefficients and the
dispersion parameter is
̂̂
=
⎛
⎞
0002424 −0001537 −0001472 −0001042 −0001033 −0001537 0000033
−0001537 0003249 0001692 −0000080 −0000286 0000824 0000022
−0001472 0001692 0006073 −0000216 −0000537 0001077 0000319
−0001042 −0000080 −0000216 0002859 0000217 −0000018 0000008
⎜
⎝ −0001033 −0000286 −0000537 0000217 0003038 0000688 0000200 ⎟
⎠
−0001537 0000824 0001077 −0000018 0000688 0006785 −0000019
0000033 0000022 0000319 0000008 0000200 −0000019 0020850
The Type 3 analysis gives the following results:
Source DF Chi-square Pr>Chi-sq
Gender ∗ Age 2 6546