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

Mixed Poisson Models for Claim Numbers 37
Example 1.3 Assume, as above, that policyholder i has been observed during a period d i
and produced k i claims. Assuming that the annual number of claims filed by policyholder i
is Negative Binomially distributed with mean d i , the log-likelihood is
La =
n∑
k i −1
∑
i=1 j=0
n∑
n∑
lna + j + na ln a − a + k i lna + d i + ln k i + constant
The maximum likelihood estimators for a and solve
k
n∑ i −1
a La = ∑ 1
n∑
n∑
i=1 j=0
a + j + n ln a + n − a + k
lna + d i −
i
= 0
i=1
i=1
a + d i
n∑
La =− a + k
d i
i + 1 n∑
k
a + d i i = 0
i=1
i=1
i=1
These equations do not possess explicit solutions, and must be solved numerically. A
convenient choice is to use the Newton–Raphson algorithm (see Section 1.5.3). Initial
values for the parameters are obtained by the method of moments. Specifically, the moment
estimator of is simply
i=1
ˆ =
∑ n
i=1 k i
∑ n
i=1 d i
which is the maximum likelihood estimate of in the homogeneous Poisson case. For the
variance, we start from VN i = EN i + d i 2 where = V i . The empirical analogue
is given by
ˆ =
∑ n
i=1
(k i − d i 2 − d i
)
∑ n
i=1 d i 2
from which we easily deduce an estimator for a in the Negative Binomial case.
1.5.2 Properties of the Maximum Likelihood Estimators
Maximum likelihood estimators enjoy a number of convenient properties that are discussed
below. It is important to note that these are asymptotic properties, i.e. properties that hold
only as the sample size becomes infinitely large. It is impossible to say in general at what
point a sample is large enough for these properties to apply, but the majority of actuarial
applications involve large data sets so that actuaries generally trust in the large sample
properties of the maximum likelihood estimators.
Consistency
First, maximum likelihood estimators are consistent. There are several definitions of
consistency, but an intuitive version is that as the sample size gets large the estimator is
increasingly likely to fall within a small region around the true value of the parameter. This