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

Bonus-Malus Scales 195
Now, we have to minimize (4.18) that can be rewritten as
[
] [
]
E exp−c − r L = E expcr L − r exp
L
× expc exp ( E [ ln E exp−cL ])
Invoking Jensen’s inequality yields
[
E exp−c − r L )] (
≥ exp cE [ ] )
r L − r exp
L
} {{ }
=1
× expc exp
[
= E
( [
E
exp−c − r exp
L
])
ln E exp−cL
]
which ends the proof.
□
Let us now compute the quantities in (4.19). Firstly,
[
]
∣
Eexp−cL = l =E Eexp−cL = l ∣L = l
= ∑ k
Eexp−cL = l = k Pr = k L = l
= ∑ k
∫ +
0
exp−c PrL = l = = kw k
dF
PrL = l = k
× Pr = kL= l
PrL = l
∫ + ∑k w k exp−c
0 l k dF
= ∫ +
(4.20)
∑k w k
0 l k dF
and secondly
[
]
E ln Eexp−cL
(4.21)
=
=
s∑
PrL = l ln Eexp−cL = l
l=0
( ∑ ∫
s∑
+
)
k w k exp−c
0 l k dF
PrL = l ln
(4.22)
PrL = l
l=0