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

Credibility Models for Claim Counts 129
Proposition 3.1 indicates that the best approximation (with respect to the mean squared
error) to T+1 given X 1 X 2 X T is EX T+1 X 1 X 2 X T , that is, the posterior
expectation of X T+1 given X 1 X 2 X T (also called the predictive mean). The posterior
distribution of X T+1 is then obtained by conditioning on past claims history.
To calculate the predictive mean one needs a conditional distribution of losses given the
parameter of interest (often the conditional mean) and a prior distribution of the parameter
of interest.
3.3.2 Predictive Distribution
Let us now come back to the credibility model of Definition 3.1. The conditional distribution
of N iTi +1 given N i1 = k 1 N iTi = k Ti
is called the predictive distribution. It tells the
actuary what the next year number of claims might be given the information contained in
past claims history. It is the relevant distribution for risk analysis, management and decision
making.
In our case, we have
PrN iTi +1 = kN i1 = k 1 N iTi = k Ti
∫ (
∏Ti
)
0 t=1 PrN it = k t i = PrN iTi +1 = k i = dF
=
(
∏Ti
)
t=1 PrN it = k t i = dF
=
∫
∫
0
exp
(
−
0 i• ) (
k • exp − iTi +1 ) iT i +1 k
dF
k!
∫
exp ( −
0 i• )
k • dF
Now, the posterior distribution of i given past claims history N i1 = k 1 N iTi = k Ti
given by
(
∏ Ti
t=1 exp ( − it ) )
it k it
dF
k it !
(
∫ ∏ Ti
0 t=1 exp ( − it ) )
it k it
dF
k it !
= exp ( − i• ) k • dF
∫
exp ( −
0 i• ) k • dF
is
Hence,
PrN iTi +1 = kN i1 = k 1 N iTi = k Ti
∫
= exp ( − iTi +1 ) iT i +1 k
dF
0
k! k •
where F ·k • is the conditional distribution function of i given N i• = k. The predictive
distribution thus appears as a Poisson mixture distribution, where the mixing is with respect