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

Efficiency and Bonus Hunger 231
Let C ik be the cost of the kth claim reported by policyholder i; we assume that the
individual claim costs C i1 C i2 are independent and identically distributed. Each C ik
conforms to the Gamma law with mean
(
)
p∑
i = EC ik x i = exp 0 + j x ij (5.5)
and variance VC ik x i = 2 i
/. Note that here we use an exponential link between the linear
predictor 0 + ∑ p
j=1 jx ij and the expected value i . For theoretical reasons, a reciprocal
link function is sometimes preferable (but destroys the nice multiplicative structure of the
resulting price list).
Let n i be the number of claims reported by policyholder i, and let c i1 c i2 c ini be the
corresponding claim costs. The likelihood associated with the observations is
= ∏
n
∏ i
in i >0 k=1
The corresponding log-likelihood is given by
in i >0
j=1
( ( ) 1
(
cik
exp − c ) )
ik 1
i i c ik
L= ln
= ∑
(
) )
n p∑ ∑ i
(−n i ln + n i ln − 0 − j x ij + ln c ik − n
∑ i
c
ik
i
+ constant
The likelihood equations are given by
L= 0 ⇔ ∑
j
in i >0
j=1
k=1
(
x ij n i − c )
i•
= 0
i
for j = 1p, where c i• = ∑ n i
k=1 c ik is the total cost of the standard claims reported by
policyholder i. The maximum likelihood estimators are obtained with the help of Newton-
Raphson techniques. The estimation of can be performed by maximum likelihood as in
Chapter 2, or it can be obtained from the Pearson- or deviance-based dispersion statistic.
Remark 5.2 Often, only the total claim amount C i• is available, and not the individual C ik s.
In such a case, it is convenient to work with the mean claim amount C i = C i• /n i , where
n i is the number of claims reported by policyholder i. Considering the Gamma likelihood
equations, this is not restrictive since only the total claim amount is needed. Specifically,
Formula (1.36) shows that the Gamma distributions are closed under convolution in some
particular cases. In the new parameterization of the Gamma family used in the present
chapter, the moment generating function of C ik is
(
Mt = 1 − t ) −
i
k=1