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

44 Actuarial Modelling of Claim Counts
where ̂ n and ̂ n are the maximum likelihood estimators in each model based on the sample
k 1 k n and f k is defined as in (1.46).
If both models are strictly non-nested (so that standard likelihood ratio tests do not apply)
then under H 0
LR̂ n ̂ n
̂ n
√ n
is approximately or0 1 distributed
where
̂ n = 1 n
(
n∑
i=1
ln pk î n
qk i ̂ n
) 2
−
(
1
n
n∑
i=1
)
ln pk 2
î n
qk i ̂ n
This provides a very simple test for model selection. Specifically, the actuary chooses a
critical value z from the or0 1 distribution for some significance level . If the value
of the test statistic is higher than z then he rejects the null hypothesis that the models are
equivalent in favour of p· being better than q·. If the test statistic is smaller than −z
then he rejects the null hypothesis in favour of q· being better than p·. Finally, if the
test statistic is between −z and z then we cannot discriminate between the two competing
models given the data.
The test statistic can be adjusted if the competing models do not have the same number
of parameters, i.e. dim ≠ dim (which is not the case in this chapter).
1.6 Numerical Illustration
Here, we consider a Belgian motor third party liability insurance portfolio observed during
the year 1997 (henceforth referred to as Portfolio A). The observed claim distribution is
given in Table 1.1. A thorough description of this portfolio is deferred to Section 2.2.
We see from Table 1.1 that the total exposure is not equal to the number of policies due
to the fact that some policies have not been in force during the full observation period (12
months). Some of them have been cancelled before the end of the observation period. Others
have been written after the start of the observation period.
Let us now fit the observations to the Poisson, the Negative Binomial, the Poisson-Inverse
Gaussian and the Poisson-LogNormal distributions. The results are summarized below:
Table 1.1 Observed claim distribution in Portfolio A.
Number of claims Number of policies Total exposure (in years)
0 12962 10 54594
1 1369 1 18713
2 157 13466
3 14 1108
4 3 252
Total 14505 11 88135