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

Credibility Models for Claim Counts 139
Table 3.7
Lower s
and upper s bounds in (3.8) for s = 2to5.
s s
s
2 2 1
b 1
3
4
2 2
( ) b1 − 3
2 b − 1 2
1 b − 1 b − 1 2 + + 2
2 b3
b − 1 2 + 2
(
1 − )
1 − r −
r− 4 r + − r + ( )
1 − r −
3 − b 4
r+ 4 p 2
− r + − r 1 + p
− 2 − b 2 b 4
1
5 q + t 5 + + q −t 5 −
p + z 5 + + p −z 5 − + 1 − p + − p − b 5
Now, let X ∈ s 0 b . The sth canonical moment of X, denoted as c s X, is given
by
c s X = EXs − s
s − s
Thus, c s X is simply the position of the sth moment of X relative to its possible range.
Now, c s X gives a good indication of the ‘position’ of X in s 0 b with respect to
the extrema X s
min
and Xs max. Therefore, we might expect that a satisfactory approximation
of X ∈ s 0 b is furnished by a convex combination of the stochastic extrema in
s−1 0 b with weights depending on the s − 1th canonical moment c s−1 X, i.e. we
use the mixture
˜X s =
{
X
s−1
min
X s−1
max
with probability 1 − c s−1 X
with probability c s−1 X
(3.9)
in order to approximate X. In the following, we refer to ˜X s as the sth canonical approximation
of X. It is easily seen that ˜X s ∈ s 0 b .
The Unimodal Case
A purely discrete approximation to the risk parameter i causes problems when an experience
rating plan has to be designed, as shown in Walhin & Paris (1999). The a posteriori
corrections obtained with discrete i s exhibit plateaus before and after sudden jumps,
which is commercially unacceptable. When F only has a few support points, a ‘block’
structure is clearly apparent for the credibility coefficients, each block with almost constant
a posteriori corrections corresponding to one support point of F . The policyholder is
transferred from smaller to larger mass points as more claims are filed. In order to avoid
this, we need a smooth risk distribution. Therefore, we would like to have simple continuous
approximations to the risk parameter. This can be done in the unimodal case, as shown
below.
A situation of practical interest in actuarial sciences is when the random variables under
consideration are known to have a unimodal distribution with a given mode m, together
with the fixed moments 1 2 s−1 . The Gamma, LogNormal and Inverse Gaussian