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

Credibility Models for Claim Counts 131
3.3.4 Poisson-Gamma Credibility Model
Gamma Distribution for the Random Effects
Let us assume that i ∼ ama a with probability density function (1.35). The joint
probability mass function of the random vector N i = N i1 N i2 N iTi is given by (2.19).
The joint probability density of the random vector i N i1 N i2 N iTi is given by
T
∏ i
t=1
( ) ( ) kit
i it
exp − i it
k it !
(
T
∑ i
∝ exp − i
t=1
1
a aa a−1
i
exp−a i
)
∑ Ti
t=1
it
k it+a−1
i exp−a i (3.3)
A Posteriori Distribution of Random Effects
In Section 2.5.1, we established that in a two-period model, the a posteriori distribution of
i remained Gamma, with updated parameters. Let us now extend the result to a multiperiod
setting.
Now, the conditional distribution of i given the past claims frequencies N it = k it , t =
1 2T i , is obtained from (2.19) and (3.3). This gives
(
exp
(− i
∫ +
0
exp
(
−
a + ∑ T i
t=1 it
(
a + ∑ T i
t=1 it
( ))
T
∑ i
= exp
(− i a + it
t=1
))
a+∑T i
t=1 k it−1
i
))
a+∑T i
t=1 k it−1
d
a+∑T i
t=1 k it−1
i
(
a + ∑ T i
t=1 it
) a+
∑ Ti
t=1 k it
(
a + ∑ )
T i
t=1 k it
Coming back to (1.34), we recognize a Gamma probability density function. Specifically,
we thus have that
The correction coefficient is given by
i N i1 N i2 N iTi ∼ am ( a + N i• a+ i•
)
(3.4)
E i N i1 N i2 N iTi = a + N i•
a + i•
which clearly increases in the past claims N i• . The variance of i given past claim history
is given by
V i N i1 N i2 N iTi =
a + N i•
a + i• 2
The expected number of claims in year T i + 1 given past claims history is given by
a + N
EN iTi +1N i1 N i2 N iTi = iTi +1E i N i1 N i2 N iTi = i•
iTi +1
a + i•