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

84 Actuarial Modelling of Claim Counts
The maximum likelihood estimator for and a solve
(
)
n∑
La= a + k
˜x i k i − i
i = 0
a + i
i=1
Note that these equations are similar to the ones obtained in the Poisson case except that i
is now replaced with i a + k i /a + i .
Remark 2.4 Let us give an intuitive explanation for the ratio a + k i /a + i involved
in the Negative Binomial likelihood equations. The joint probability density function of
N i i equals
( ) ki
i i 1
exp− i i
k i ! a aa a−1
i
exp−a i
∝ exp − i i k i+a−1
i exp−a i
The probability density function of i given N i = k i is
exp − i a + i a+k i−1
i
∫ +
0
exp − a + i a+k i−1
d
= exp − i a + i a+k a +
i−1 i a+k i
i
a + k i
so that i given N i = k i follows the am a + k i a+ i distribution. Therefore,
E i N i = k i = a + k i
a + i
and the maximum likelihood estimators in the Negative Binomial regression model solve
n∑
i=1
)
x i
(k i − i E i N i = k i = 0
Compared to the Poisson likelihood equations, the predicted expected claim number i is
replaced with its update i E i N i = k i based on the information contained in the number
k i of claims filed by policyholder i.
As already shown in Section 2.3.7, it is possible to solve the Negative Binomial likelihood
equations with the help of the Newton–Raphson iterative procedure. Starting values for the
Newton–Raphson iterative procedure are usually obtained as follows: The Poisson maximum
likelihood estimator ̂ is known to be consistent, so that we keep it as a reasonable starting
value. We still have to find an initial estimate for = V i = 1/a. So, we first compute
the variance of N i which is given by
VN i = EN i + ( d i expscore i ) 2