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

Risk Classification 83
T 2 =
∑ n
i=1
√
∑n
i=1
(k i −̂ i 2 − k i
)
(
k i −̂ i 2 − k i
) 2
and
T 3 = √
1
n
∑ n
i=1
(k i −̂ i 2 − k i
)
∑
(
) 2
√
n ∑n
i=1̂ −2
i k i −̂ i 2 − k i i=1̂ 2 i
All these test statistics are or0 1 distributed. For Portfolio A, T 1 = 918, T 2 = 613 and
T 3 = 438. All the p-values are less than 10 −4 leading to the rejection of the null hypothesis
(and thus the Poisson model) in favour of the mixed Poisson model.
2.5 Negative Binomial Regression Model
2.5.1 Likelihood Equations
Overdispersion is taken into account by the inclusion of a random effect, representing
an unknown relative risk level. More precisely, assume that 1 n are independent
ama a distributed random variables, i.e. the common probability density function
of the i s is given by (1.35). In this case, E i = 1 and V i = 1/a.
Note that the assumptions made about the i s are rather strong. Their common
distribution means that the effect of hidden variables does not depend on observable
ones.
Conditional on the observable characteristics summarized in the vector x i and on the
random effect i = , the annual claim number caused by policyholder i conforms to the
oi i law. In other words, i is the expected claim frequency for policyholder i (based
on x i ) and is the relative risk level for this policyholder (if