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

82 Actuarial Modelling of Claim Counts
0.8
Overdispersion of the data
Empirical variances
0.6
0.4
0.2
0.0
0.0 0.1 0.2 0.3 0.4
Empirical means
Figure 2.9 Mean–Variance pairs for the risk classes, Portfolio A.
A quadratic curve (without intercept) has been fitted to the mean–variance couples by
weighted least-squares (the weights being the exposures of the risk classes). The high value
of the R 2 coefficient (86.41 %) supports the quality of the fit. This shows that equation (2.11)
is supported by the data, and provides empirical evidence for a mixed Poisson model.
2.4.6 Testing for Overdispersion
The graphical test of the previous section is an easy way of detecting overdispersion. But
many statistical tests for the overdispersion assumption have been developed in the literature.
Testing for overdispersion can be done by testing for the Poisson distribution against a mixed
Poisson model. One problem with standard specification tests (such as likelihood ratio tests)
occurs when the null hypothesis is on the boundary of the parameter space, as explained in
Chapter 1. When a parameter is bounded by the H 0 hypothesis, the estimate is also bounded
and the asymptotic Normality of the maximum likelihood estimator no longer holds under
H 0 . Consequently, a correction must be made.
Alternatively, testing for overdispersion can be based on the variance function. According
to (2.11), the variance function of a heterogeneity model is of the form:
VN i = i + 2 i (2.12)
with = V i being the variance of the random effect. Therefore, we have to test the null
hypothesis H 0 : = 0 against H 1 : >0. The following score statistics can be used to test the
Poisson distribution against heterogeneity models with a variance function of the form (2.12):
T 1 =
∑ n
i=1
(k i −̂ i 2 − k i
)
√
2 ∑ n
i=1̂ 2 i