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

136 Actuarial Modelling of Claim Counts
PrN iTi +1 = iN i• = k = PrN iT i +1 = i and N i• = k
PrN i• = k
∑ q
j=1
=
PrN iT i +1 = i i = j PrN i• = k i = j p j
∑ q
j=1 PrN i• = k i = j p j
=
q∑
exp− iTi +1 j iT i +1 j i p j exp− i• j j
k
∑
i! q
l=1 p
l exp− i• l l
k
j=1
Hence, given N i• = k, the law of N iTi +1 appears as a discrete Poisson mixture with modified
weights
˜p j k =
p j exp− i• j j
k
∑ q
l=1 p
l exp− i• l l
k
The a posteriori expectation of N iTi +1 is then
EN iTi +1N i• = k = iTi +1
The posterior mean of i given N i• = k is then given by
∑ q∑
q
j=1
j˜p j k = p j exp− i• j k+1
j
iTi +1 ∑ q
l=1 p (3.6)
l exp− i• l l
k
j=1
E i N • = k =
∑ q
j=1 p j exp− i• j k+1
j
∑ q
l=1 p (3.7)
l exp− i• l l
k
Compared to (3.2), the integrals now reduce to sums over the q components of the discrete
mixture.
Even if the reality of the insurance portfolio is a discrete mixture (with a i specific to
policyholder i, resulting in q = n), this model is not convenient since it involves a large
number of parameters. The discrete Poisson mixture nevertheless deserves interest as an
approximation of more general Poisson mixtures, as discussed below.
3.3.8 Discrete Approximations for the Heterogeneous Component
Moment Spaces
Apart from the Poisson-Gamma case, the computation of the conditional expectation giving
the credibility premium requires numerical integration. A convenient alternative is to
approximate the mixing distribution by a suitable discrete analogue sharing the same sequence
of moments. We are then back to the discrete Poisson mixture credibility model studied
above, and we benefit from the easy-to-compute formulas (3.6)–(3.7) valid in this case.
The discrete approximations to i are based on the knowledge of its support, 0b say,
with b possibly infinite, and its first few moments 1 2 In general, let us denote by
s 0 b the class of all the random variables X with support in 0band with prescribed
first s −1 moments EX k = k , k = 1 2s−1. In the literature, s 0 b is referred
to as a moment space. Properly speaking, it is a class of distribution functions rather than a
class of random variables. Classical problems related to moment spaces are for instance the