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

Bonus-Malus Scales 183
(i) set A0 = 1
(ii) compute for l = 0 1s− 1
(iii) set
Al + 1 =
(
1
Al −
p −1
F l = Al
As
)
l∑
Al − yp y
y=0
for l = 0 1s
This recursive formula is computationally efficient, and easy to implement.
4.4.4 Convergence to the Stationary Distribution
Geometric Bound for the Speed of Convergence
What matters for a unique limit to exist is that one can find a positive integer n 0 such
that
=
min
ij∈0s
p n 0
ij > 0
In other words, n 0 is such that all the entries of P n 0 are positive, and thus it is possible
to reach any level starting from any other level in n 0 periods. Various inequalities indicate
the speed of convergence to the limit distribution . It can be shown that
p n
ij − j ≤
max
i∈0s
p n
ij −
min
i∈0s
p n
ij ≤ ( 1 − )⌊ ⌋
n
n0 −1
This inequality provides us with a geometric bound for the rate of convergence to the limit
distribution. Further bounds can be determined using concepts of matrix algebra.
Total Variation Distance
The total variation metric is often used to measure the distance to the stationary distribution
. Recall that the total variation distance between two random variables X and Y , denoted
as d TV X Y, is given by
d TV X Y =
∫ +
−
dF X t − dF Y t (4.10)
For counting random variables M and N , (4.10) obviously reduces to
d TV M N =
+∑
k=0
∣ PrM = k − PrN = k ∣ ∣
There is a close connection between d TV and the standard variation distance, which
considers the supremum of the difference between the probability masses given to some