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

264 Actuarial Modelling of Claim Counts
6.2.7 Estimation of the Variances and Covariances
The formulas derived above for VNi•
mat bod
mat
, VNi•
and CNi•
estimates for the parameters mat 2 , 2 bod and bm:
∑
(
n
(k ) )
mat mat 2
̂ mat 2 i=1 i•
− ̂ i• − k
mat
i•
=
̂ 2 bod = ∑ n
i=1
̂ bm =
∑ n
i=1
∑ n
i=1 (̂mat i•
( (k
bod
i•
(
) 2
) )
bod 2
− ̂ − k
bod
i•
∑ n
i=1 (̂bod i•
ki•
mat
∑ n
i=1
) 2
)(
mat
− ̂
i•
̂
mat
i•
k bod
i•
bod ̂ i•
i•
Nbod i•
)
bod
− ̂
i•
suggest the following
that are consistent in the random effects model.
The parameters mat 2 , 2 bod and bm have been estimated above on aggregate data, giving
the estimators ̂ mat, 2 ̂ bod 2 and ̂ bm. Alternatively, these parameters could be estimated from
individual data as follows:
∑ n ∑
(
Ti
(k ) )
mat mat 2
i=1 t=1 it
− ̂ it − k
mat
it
˜ 2 mat =
∑ n ∑ Ti
˜
bod 2 = i=1 t=1
˜ bm =
∑ n
i=1
∑ Ti
t=1
∑ n
i=1
∑ Ti
t=1
( (k
bod
it
−
∑ n
i=1
∑ Ti
t=1
(
k
mat
it
−
∑ n
i=1
∑ Ti
t=1
(̂mat it
̂
bod
it
(̂bod it
̂
mat
it
̂
mat
it
) 2
) )
2
− k
bod
) 2
it
)(
k
bod
−
it
̂ bod
it
)
bod ̂ it
The estimators ̂ 2 mat, ̂ 2 bod and ̂ bm are preferred over ˜ 2 mat, ˜ 2 bod and ˜ bm, respectively, since
the variances of the former are smaller. As shown by Pinquet ET AL. (2001), the condition
0 < ̂ 2 mat < ˜ 2 mat is necessary for the introduction of dynamic random effects.
6.2.8 Linear Credibility Premiums
Denote as
mat
iT i +1 = d iT i +1 expscore mat
iT i +1 and as bod
iT i +1 = d iT i +1 expscore bod
iT i +1
the expected claim frequencies for policyholder i in period T i + 1. The best linear predictor
∑
T i
c mat
i0
+
t=1
c mat/mat
it
N mat
∑
T i
it
+
t=1
c bod/mat
it
N bod
it