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

262 Actuarial Modelling of Claim Counts
where f · · is the bivariate probability density function of the couple of random effects
i
mat i
bod . The joint probability density function of i
mat , i
bod , Nit
mat , Nit
bod , t = 1T i
is given by
exp− mat
i
mat
i•
− bod i
bod
i•
mat i
kmat i•
∏ Ti
bod
i
kbod i•
so that the conditional probability density function of mat
i
∫
0
exp− mat
∫
i
mat
i•
− bod i
bod
i•
mat i
kmat i•
bod
i
exp− 0 1 mat
i•
The posterior distribution of i
mat
in Chapter 3.
− 2 bod
i•
i
bod
t=1 mat it
kmat it bod
it
kbod it
∏ Ti
t=1 kmat it !kit bod !
i
bod
f mat
i
bod
i
given past claims history is
kbod i• f i
mat i
bod
(6.1)
1 kmat i• 2 kbod i• f 1 2 d 1 d 2
then allows for a posteriori corrections, as explained
6.2.4 Summary of Past Claims Histories
Denote as
∑
T i
N bod
i•
= N bod
it
t=1
and N mat
∑
T i
i•
= N mat
it
t=1
the total claim numbers of each category caused by policyholder i during the observation
period. Since the random effects i
mat and i
bod do not vary with time, Ni•
mat and Ni•
bod are
sufficient summaries of past claims histories (in the sense that the posterior distributions of
i
mat and i
bod as well as the predictive distributions of NiT mat
bod
i +1
and NiT i +1
only depend on
Ni•
mat and Ni•
bod ; see (6.1) where past claims histories enter through kmat
i•
and ki• bod).
Clearly,
mat
i•
It is then easy to see that given mat
i
and that given bod
i
= bod
i
= EN mat and bod
i•
= mat
i
N mat
i•
N bod
i•
i•
∼ oi ( mat
i•
∼ oi ( bod
i•
= EN bod
i•
mat i
bod i
)
)
invoking the conditional independence of the annual claim numbers of each category and the
stability of the Poisson family under convolution. Therefore, Ni•
mat and Ni•
bod are both mixed
Poisson distributed.