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

Actuarial Analysis of the French Bonus-Malus System 331
• the probability mass function of S is denoted as
gs = PrS 1 = s 1 S k = s k
• the probability mass function of X is denoted as
fx = PrX 1 = x 1 X k = x k
• the difference between vectors has to be understood componentwise, that is,
s − x = s 1 − x 1 s k − x k T
• and, finally,
(
s∑ ∑ s1
fx = ···
s k
∑
x≠0
x 1 =0 x k =0
)
fx 1 x k − f00
Multivariate Panjer Algorithm
We are now ready to state and prove the following result.
Property 9.1 Let S be as in (9.3) with X i = X i1 X ik T , i = 1 2, independent and
identically distributed, arithmetic and independent of N . Furthermore, we assume that N
belongs to Panjer’s class, i.e. its probability mass function satisfies (7.7). Then, if N denotes
the probability generating function of N , we have
g0 = N f0 (9.4)
(
1
s∑
gs =
a + b x )
i
gs − xfx
1 − af0 s i
s i ≥ 1 i= 1k (9.5)
x≠0
Proof
From
Let X · and S · be the probability generating functions of X and S, respectively.
∑
gs = PrN = nf ⋆n s
n=0
we get
∑
S u = PrN = n ( X u ) n
from which (9.4) follows immediately. By hypothesis, one has
n=0
n PrN = n = an − 1 PrN = n − 1 + a + b PrN = n − 1
n ≥ 1