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

Multi-Event Systems 271
Denoting as the (unknown) accident proneness of this policyholder, the conditional
probability mass function of N is given by
PrN = j = = k = exp− k − k j
j = 0 1 2
j!
The risk profile of the portfolio is described by the distribution function F of and we
assume that E = 1. Since represents the residual effect of unobserved characteristics, it
seems reasonable to assume that and are mutually independent. Hence, the unconditional
probability mass function of N is given by
PrN = j = ∑ k
∫ +
w k PrN = j = = k dF
0
j = 0 1
We distinguish among m different types of claim reported by the policyholder. Each type
of claim induces a specific penalty for the policyholder. For instance, one could think of
• claims with bodily injuries and claims with material damage only (m = 2)
• claims with partial liability and claims with full liability (m = 2)
• introducing claim severities (for instance, claims with amount less than E1000, between
E1000 and E10 000, and claims above E10 000, so that m = 3). In this case, we have to
assume that claim severities and claim frequencies are mutually independent.
Here, we will assume that the claims are classified according to a multinomial scheme.
Specifically, each time a claim is reported, it is classified in one of the m possible categories,
with probabilities q 1 q m . Let us denote as N i the number of claims of type i. Then, the
random vector N 1 N m is Multinomially distributed, with probability mass function
PrN 1 = k 1 N m = k m =
{
n!
k 1 !···k m ! qk 1
1 ···qk m m
0 otherwise
if k 1 +···+k m = n
where n is the total number of claims.
Each of the m components separately has a Binomial distribution with parameters n and
q i , for the appropriate value of the subscript i, that is, N i ∼ inn q i . Because of the
constraint that the sum of the components is n, that is, N 1 +···+N m = n, they are negatively
correlated.
The expected value is EN i = nq i . The covariance matrix is as follows: Each diagonal
entry is the variance of a Binomially distributed random variable, and is therefore
VN i = nq i 1 − q i . The off-diagonal entries are the covariances. These are
CN i N j =−nq i q j for i, j distinct. This is a m × m nonnegative-definite matrix of rank
m − 1.
We will use the following result.
Property 6.1 Let us assume that the total number of claims N is oi distributed.
Assume that the N claims may be classified into m categories, according to a multinomial
partitioning scheme with probabilities q 1 q m . Let N i represent the number of claims of