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

142 Actuarial Modelling of Claim Counts
When the risk parameter i is known to be unimodal (with mode m, say) then the
improved extremal distributions of Tables 3.8–3.9 can be used in lieu of those coming from
Tables 3.5–3.6. This amounts to evaluating the numerator and denominator of (3.2), taking
for the structure function F a discrete mixture of uniform distributions. This provides an
easy-to-compute approximation for (3.2) based on incomplete Gamma functions, as can be
seen from the following example.
Example 3.1 Assume that i has support + , mode m and moments 1 and 2 . Then, we
can use the approximation Mk ≈ M 3
mink where
M 3
min k = 3 2 + 2 1 m − m 2 − 4 2 1
1
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
m
∫ m
m − 2
+
1 2
1
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
c − c
0
exp− k d
∫ c
c
exp− k d
with c and c defined as
and
(
c = min m 3 )
2 − 2m 1
2 1 − m
(
c = max m 3 )
2 − 2m 1
2 1 − m
The value of M 3
mink is easily obtained from the incomplete Gamma function. Specifically,
M 3
min k = 3 2 + 2 1 m − m 2 − 4 2 1
k!
k + 1 m
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
mk+1 m − 2
+
1 2
k! ( )
k + 1 c− k + 1c
m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
c − c k+1
Example 3.2 Now, if the support of i is known to be contained in 0b then we can use
the approximation Mk ≈ Mmaxk 3 where
M 3
max k = 3 2 + 2 1 m − m 2 − 4 2 1
1
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1
b − m
b + m − 2
+
1 2
1
b + m − 2 1 2 + 3 2 + 2 1 m − m 2 − 4 2 1 d − d
∫ b
m
∫ d
d
exp− k d
exp− k d
with d and d defined as
(
d = min m 2b )
1 + 2 1 m − bm − 3 2
b + m − 2 1