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

68 Actuarial Modelling of Claim Counts
in (1.47). Let us define ˜x i as the vector x i of explanatory variables for policyholder i
supplemented with a unit first component, that is, ˜x i = 1 x T i T . Then, considering (2.3)–
(2.4), U is given by
U =
n∑
˜x i k i − i (2.5)
i=1
in the Poisson regression model. Let H be the Hessian matrix of L defined in (1.50).
Specifically, H is given by
H =−
n∑
˜x i˜x T i i (2.6)
i=1
in the Poisson regression model. The maximum likelihood estimator ̂ j of the parameters
j then solves U = 0.
The approach used to solve the likelihood equations is the Newton–Raphson algorithm
(1.51). Starting from an appropriate ̂ 0 , the Newton–Raphson algorithm is based on the
following iteration
̂r+1 = ̂
)
r −1
− H
(̂r U
(̂r)
(2.7)
= ̂ r +
( ) n∑ −1
˜x i˜x T ̂
r n∑ (
i i ˜x i k i − ̂ ) r
i
i=1
i=1
for r = 0 1 2, where ̂ i
r
= di exp˜x T i ̂ r . Appropriate starting values are given by
̂ 0
0
= ln
1
n
n∑
k i and ̂
0
j = 0 for j = 1p.
i=1
Note that these starting values are equal to the values of the regression coefficients when no
segmentation is in force. Therefore, final values close to the starting ones indicate that the
portfolio is quite homogeneous.
Remark 2.2 (Iterative Least-Squares) It is possible to interpret the Newton–Raphson
approach (2.7) in terms of iterative least-squares. Specifically, it is possible to rewrite
the iterative algorithm (2.7) in the Poisson model in such a way that ̂r+1
appears as the maximum likelihood estimator in a linear model with adjusted dependent
variables. Fitting the Poisson regression model by maximum likelihood thus boils down
to estimating the regression parameter in a sequence of linear models, with adjusted
responses and explanatory variables. This is particularly interesting since the numerical
aspects of estimation in a linear model are well-known and have been optimized for
decades.