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

184 Actuarial Modelling of Claim Counts
random events. Specifically, given two random variables X and Y , d TV X Y can be
represented as
∣
d TV X Y = 2 sup ∣ PrX ∈ A − PrY ∈ A∣ (4.11)
A
Selection of the Initial Level
The main objective of a bonus-malus system is to correct the inadequacies of a priori rating
by separating the good from the bad drivers. This separation process should proceed as fast
as possible; the time needed to achieve this operation is the time needed to reach stationarity.
A convenient way to select the initial level has been suggested by Bonsdorff (1992). The
idea is to select it in order to minimize the time needed to reach stationarity. It relies on
the total variation distance d TV between the nth transient distribution starting from level l 1 ,
i.e. p n
l 1 l 2
l 2 = 0 1s, and the stationary distribution l2
l 2 = 0 1s,
computed as
d TV l 1 n=
s∑
l 2 =0
p n
l 1 l 2
− l2
it measures the degree of convergence of the system after n transitions. Of course,
lim
n→+ d TV l 1 n= 0 for all l 1 and
The convergence to is essentially controlled by the second largest eigenvalue 1 .
For any > 1 , there exists an a