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

Risk Classification 71
2.3.11 Deviance
Let ̂ be the model likelihood, i.e.
̂ =
n∏
i=1
exp−̂ i ̂ k i
i
k i !
Note that the maximal value of ↦→ exp− k /k! is obtained for = k. Therefore, the
Poisson likelihood is maximum with expected claim frequencies equal to the observed
number of claims. The maximal likelihood possible under the Poisson assumption is then
k =
n∏
i=1
exp−k i kk i
i
k i !
This is the likelihood of the saturated model, predicting the observed number of claims for
each insured driver (there are thus as many parameters as observations). This model just
replicates the observed data.
The deviance Dk̂ is defined as the likelihood ratio test statistic for the current model
against the saturated model, that is,
Dk̂ =−2ln ̂ (
)
k = 2 ln k − ln ̂
( ) (
∏ n
∏ n
= 2ln
= 2
i=1
(
n∑
i=1
exp−k kk i
i
i k i !
k i ln k i
̂i
− k i −̂ i
− 2ln
)
i=1
)
exp−̂ ̂ k
i
i
i k i !
where y ln y = 0 for y = 0 by convention. It measures the distance of the model likelihood
to the saturated model replicating the observed data. The smaller the deviance, the better the
current model.
When an intercept 0 is included in the linear predictor, (2.3) allows us to simplify the
deviance as
Dk̂ = 2
n∑
i=1
k i ln k i
̂i
Provided the data have been grouped as much as possible and the model is correct, D is
approximately 2 n−dim
distributed (where n is now the number of classes in the portfolio).
The model is considered as inappropriate if D obs is ‘too large’, that is, if
D obs > 2 n−dim1−