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

230 Actuarial Modelling of Claim Counts
Theoretically, any distribution function F can be used; in practice, we often take for F
the Normal or Logistic distribution function. For instance, the logistic regression model is
specified as
q
ln i x i
p∑
1 − q i x i = 0 + j x ij ⇔ q i =
exp ( 0 + ∑ p
j=1 )
jx ij
1 + exp ( 0 + ∑ p
j=1 )
jx ij
j=1
The SAS R /STAT procedure GENMOD can be used to perform this analysis. As in the
Poisson regression case, all the explanatory variables described in Section 5.1.6 are excluded
from the model. The estimated probability that policyholder i does not report any large
claim is 99.99013 %. Hence, the corresponding large claim frequency is − ln 09999013 =
000987 %, which is not too far from the estimated large
i obtained using Poisson regression.
5.2.5 Modelling the Costs of Moderate Claims
Different models can be used to describe the behaviour of the moderate claims (i.e. claims
with an incurred cost less than E100 000) as a function of the observable characteristics
of the policyholder; including Gamma, Inverse Gaussian and LogNormal distributions. We
briefly review these three regression models next.
Gamma Distribution
Here we use a new parameterization of the Gamma probability density function (1.34).
Specifically, we use the mean as parameter, together with a parameter related to the variation
coefficient. The probability density function with the new parameters = / and = is
then given by
fy = 1
( ) y (
exp − y ) 1
y (5.4)
If Y has probability density function (5.4), then the first moments are given by
EY = and VY = 2
so that the variance is proportional to the square of the mean. Gamma regression assumes
a coefficient of variation constantly equal to −1/2 . Thus it allows for heteroscedasticity
(since the variance is proportional to the square of the mean, and is no more constant as in
Gaussian regression models). Ideally, the Gamma regression model is best used with positive
observations having a constant coefficient of variation. However, the model is robust to wide
deviations from the latter assumption.
The parameter controls the shape of the probability density function. Specifically, (i) if
0