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

102 Actuarial Modelling of Claim Counts
The variance-covariance matrix of the N it s in the Poisson model with serial independence is
given by
⎛
⎞
i1 0 ··· 0
0 i2 ··· 0
A i = ⎜
⎝
⎟
⎠
0 0 ··· iTi
In fact this matrix does not take the overdispersion and the serial dependence of the data
into account. Noting that
EN i = A i X i
it is possible to transform (2.16) in order to let A i explicitly appear in the likelihood equations.
This gives
n∑
i=1
( )
T
EN i A ( −1
i n i − EN i ) = 0 (2.17)
The principle of Generalized Estimating Equations (GEE) is to find a suitable variancecovariance
matrix to insert in Equation (2.17) instead of A i based on serial independence.
This matrix should take the overdispersion and the serial dependence into account. A possible
form of this matrix could be
V i = A 1/2
i R i A i
1/2
where the ‘working’ correlation matrix R i takes the serial dependence between the
components of N i into account and depends on a parameter . The overdispersion is also
taken into account as VN it = it exceeds EN it = it provided >1.
The idea behind the GEE method is then to replace A i by V i in (2.17) and to compute the
estimator of as the solution of
n∑
i=1
( )
T
EN i V ( −1
i n i − EN i ) = 0 (2.18)
The resulting estimator is consistent whatever the choice of the matrix R i but the precision
will be much better if R i is close to the true correlation matrix of N i . Equation (2.18)
is solved thanks to a modified version of the Fisher scoring method for and a moment
estimation for and . The iterative procedure is as follows:
1. Compute an initial estimate of assuming independence.
2. Compute the current ‘working’ correlation matrix based on standardized residuals, current
and the assumed structure of R i .
3. Estimate the covariance matrix V i .
4. Update .
Note that GEE is not a likelihood based method of estimation, so that inferences based on
likelihoods are not possible in this case.