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

Transient Maximum Accuracy Criterion 309
Cn
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0 1 2 3 4 5
Age of policy
Figure 8.2 Convergence of the transient policyholders’ distributions to the steady state distribution
for the system −1/top.
distribution of the policyholders inside the scale. This comes from the fact that the transient
distributions do not depend on the initial distribution for the −1/top scale.
This somewhat surprising situation can be explained as follows. The distribution of the
policyholders in the −1/top scale after one year is given by
⎛
⎜
⎝
p 1
0
⎞
p 1
1
p 1
2
p 1
3
p 1
4
⎟
⎠
p 1
5
⎛
=
⎜
⎝
p 0
0
p 0
1
p 0
2
p 0
3
p 0
4
p 0
5
⎞T ⎛
⎞
exp− 0 0 0 0 1− exp−
exp− 0 0 0 0 1− exp−
0 exp− 0 0 0 1− exp−
0 0 exp− 0 0 1− exp−
⎟ ⎜
⎠ ⎝ 0 0 0 exp− 0 1− exp−⎟
⎠
0 0 0 0 exp− 1 − exp−
⎛
=
⎜
⎝
p 0
0 + p 0
1 exp− ⎞
⎟
⎠
p 0
2 exp−
p 0
3 exp−
p 0
4 exp−
p 0
5
exp−
1 − exp−