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

Credibility Models for Claim Counts 143
and
(
d = max m 2b )
1 + 2 1 m − bm − 3 2
b + m − 2 1
As above, this expression can be simplified with the help of the incomplete Gamma function
as
M 3
max k = 3 2 + 2 1 m − m 2 − 4 2 1
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
k! ( )
×
k + 1 b − k + 1 m
b − m k+1
b + m − 2
+
1 2
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
k! ( )
×
k + 1 d− k + 1d
d − d k+1
Example 3.3 If the third moment of i , 3 say, is known then we can use a mixture of the
two preceding approximations, with weights defined by the canonical moments as suggested
by (3.9). To this end, note that
and
⋆ 3 =
⋆ 3 =
min
X∈ 3 0bm−unim 1 2
EX 3
3
=
2 + 2 1 m − m 2 − 4 2 1
m 3
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
3
m − 2
+
1 2
c 4 − c 4
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
c − c
max
X∈ 3 0bm−unim 1 2
EX 3
3
=
2 + 2 1 m − m 2 − 4 2 1
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
b + m − 2
+
1 2
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
The approximation to the Mellin transform is then as follows
b 4 − m 4
b − m
d 4 − d 4
d − d
Mk ≈ ⋆ 3 − 3
⋆ 3 − M 3
⋆ 3
min k + 3 − ⋆ 3
⋆ 3 − ⋆ 3
M 3
max k