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

146 Actuarial Modelling of Claim Counts
Table 3.12 Values of E i N i• = k • for different combinations of observed periods T i and
number of past claims k • for a bad driver from Portfolio A (expected annual claim frequency
of 28.40 %) and for a mixture of mixed uniform approximations of i (improved 4-convex
extrema).
T i
Number of claims k •
0 1 2 3 4 5
1 78.95 % 15307 % 227.29 % 301.52 % 374.70 % 449.61 %
2 65.22 % 12650 % 187.69 % 249.57 % 310.74 % 369.83 %
3 55.55 % 10812 % 159.34 % 212.96 % 265.29 % 317.66 %
4 48.36 % 9488 % 138.01 % 185.46 % 231.41 % 277.06 %
5 42.80 % 8493 % 121.75 % 163.48 % 205.61 % 245.24 %
6 38.39 % 7709 % 109.33 % 145.45 % 184.61 % 220.87 %
7 34.79 % 7065 % 99.70 % 130.79 % 166.57 % 201.26 %
8 31.83 % 6518 % 92.05 % 119.03 % 150.92 % 184.27 %
9 29.35 % 6042 % 85.75 % 109.64 % 137.59 % 168.95 %
10 27.24 % 5622 % 80.40 % 102.06 % 126.49 % 155.15 %
Specifically, we look for c i0 and c it s, t = 1T i , such that the expected square difference
between N iTi +1 and its prediction ˆN iTi +1 is minimum, i.e. such that
c = arg min 1
c
where
⎡(
) ⎤
T 2
∑ i
1 = E ⎣ N iTi +1 − c i0 − c it N it
⎦
t=1
Alternatively, it can be shown that c also solves
c = arg min j j = 2 3
c
where
⎡(
) ⎤
T 2
∑ i
2 = E ⎣ EN iTi +1 i − c i0 − c it N it
⎦
⎡(
) ⎤
T 2
∑ i
3 = E ⎣ EN iTi +1N i1 N iTi − c i0 − c it N it
⎦
t=1
t=1
Let us show for instance that arg min c 1 = arg min c 2 . To this end, let us write
⎡
(
)) ⎤
( )
T 2
∑ i
1 = E ⎣(
N iTi +1 − EN iTi +1 i + EN iTi +1 i − c i0 − c it N it
⎦
t=1