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

66 Actuarial Modelling of Claim Counts
2.3.5 Likelihood Equations
Let k i be the number of claims filed by policyholder i during the observation period. The
likelihood associated with these observations equals
where
n∏
n∏
= PrN i = k i x i = exp− i k i
i
k i !
i=1
i=1
i = d i expscore i = expln d i + score i
The maximum likelihood estimator ̂ of maximizes : ̂ is the value of the regression
coefficients that makes the observations the most plausible.
Remark 2.1 (Grouping Data) The maximum likelihood estimators obtained for individual
or grouped data are identical. Let us prove it formally. To this end, let group be the
likelihood obtained after a grouping in risk classes. We will show below that
∝ group
In words, the likelihood group based on grouped data is proportional to the likelihood
based on individual data, so that the corresponding maximum likelihood estimates will
coincide.
Let s 1 s q be the q possible values for the score, say, and let us define
d •j =
∑
iscore i =s j
d i and k •j =
∑
iscore i =s j
k i for j = 1q
In words, d •j is the total risk exposure for risk class j (corresponding to the value s j of the
score) and k •j is the total number of claims recorded for the same risk class. Then
q∏ ∏
= exp− i k i
i
j=1
k
iscore i =s j i !
⎛ ⎞
q∏
∝ exp ⎝−
∑
i
⎠ ( ) k•j
exps j d •j
j=1
iscore i =s j
⎛
q∏
∑
= exp ⎝− exps j ⎠ ( ) k•j
exps j d •j
j=1
iscore i =s j
d i
⎞
∝
q∏
exp ( ) ( ) k•j
exps j d •j
− exps j d •j
k •j !
j=1
= group
Maximizing or group then gives the same maximum likelihood estimator ̂.