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

234 Actuarial Modelling of Claim Counts
The log-likelihood is
L 2 = ln 2
= ∑
(
− n i
2 ln22 −
in i >0
so that the likelihood equations are given by
j
L 2 = 0 ⇔
j
⇔ ∑
in i >0
∑
n
∑ i
k=1
n
∑ i
in i >0 k=1
)
c ik − i 2
+ constant
2 i 2 c ik
(
1 − c ik
i
) 2
1
c ik
= 0
(
x ij
n
i − c )
i•
= 0
i i
The estimation of 2 can be performed by maximum likelihood as in Chapter 2, or it can
be obtained from the Pearson- or deviance-based dispersion statistic.
Remark 5.4 As pointed out for the Gamma distribution in Remark 5.2, the actuary often
only has at his disposal the total claim amount C i• , and not the individual C ik s. This is
not really a problem since the likelihood equations only involve C i• . Considering (1.40),
we see that the moment generating function of the mean claim amount C i in the new
parameterization is given by
√
1 − 2 2 2 i
(
exp − n (
i
1 −
2 i
))
t
n i
which corresponds to the Inverse Gaussian distribution with parameters i and 2 /n i .Asin
the Gamma case, working with the average claim amounts is not restrictive for maximum
likelihood estimation of the regression parameters, and this situation is accounted for in
GENMOD by specifying an appropriate weight n i .
Example 5.2 (Inverse Gaussian Regression for the Moderate Claim Costs in Portfolio C)
The results of the Inverse Gaussian regression are given in Table 5.4 where the following
variables have been eliminated: Fuel (p-value of 96.29 %), Gender (p-value of 84.58 %),
Power (p-value of 78.32 %), Use (p-value of 56.40 %), Premium split (p-value of 23.04 %),
City (p-value of 975 %) and Coverage (p-value of 5.37 %). Moreover, for Agev, levels 3–5,
6–10 and > 10 have been grouped together in a class > 2. For the variable Ageph, levels
31–60 and > 60 have been grouped in a class > 30. The resulting log-likelihood is equal to
−150 21460. Type 3 analysis is presented in the following table:
Source DF Chi-square Pr>Chi-sq
Ageph 2 1056 00012
Agev 1 4187