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

332 Actuarial Modelling of Claim Counts
Multiplying on both sides of the equality by n−1
X uu i/u i X u and summing over n = 1
to +, weget
u i
u S u = a X uu i
i u S u + a + bu i
i u X u S u
i
Comparing identical powers of u on both sides of the equality gives:
s∑
s∑
s i gs = a fxs i − x i gs − x + a + b fxx i gs − x
x=0
⇔ s i gs = as i f0gs +
⇔ gs =
1
1 − af0
x≠0
x=0
s∑
fxgs − x ( )
as i + bx i
x≠0
(
s∑
fxgs − x a + b x )
i
s i
which ends the proof.
□
The Panjer formula in dimension 1 immediately follows from Property 9.1 by putting
k = 1.
Multivariate De Pril Algorithm
Another interesting problem is to compute the multivariate convolution of a random vector.
This is actually the result we will need in the following sections. Let X i = X i1 X ik T ,
i = 1 2n, be independent and identically distributed realizations of the random vector
X = X 1 X k T . We want to obtain the distribution of the random vector
( ) n∑ T
n∑
S = S 1 S k T = X i1 X ik (9.6)
i=1
i=1
The probabiliy mass function of S is the n-fold convolution f ⋆n x. The multivariate version
of De Pril’s formula provides a recursive formula to derive the distribution of S.
Property 9.2 Let X i = X i1 X ik T be independent and identically distributed
realizations of the random vector X defined on the nonnegative integers and with probability
function such that f0>0. Then, the following recursion holds :
f ⋆n 0 = f n 0
f ⋆n s = 1 ( )
s∑ n + 1
x
f0 s i − 1 f ⋆n s − xfx
i
s i ≥ 1 i= 1k
x≠0
Proof Let us introduce an auxilliary random vector W with probability mass function
h0 = 0 and
hx =
fx
1 − f0 x > 0