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

128 Actuarial Modelling of Claim Counts
This result has an important practical consequence: the Poisson credibility model A1–A2
disregards the age of the claims. The penalty induced by an old claim is strictly identical
to the one induced by a recent claim. This may of course sometimes be undesirable for
commercial purposes. We will come back to this issue in the last section of this chapter.
3.3 Credibility Formulas with a Quadratic Loss Function
3.3.1 Optimal Least-Squares Predictor
Often in applied probability, one seeks to approximate an unknown quantity by a function
of a set of related variables, by minimizing the expected squared difference between the two
items. This leads to the least-squares principle, and to the conditional expectation, as shown
in the next result.
Proposition 3.1 Let us consider a sequence of random variables X 1 X 2 X 3 and a
risk parameter . Given , the X t s are independent. The first two moments of the X t s are
assumed to be finite. Moreover, the conditional mean of the X t s is given by
t = EX t t = 1 2 3
and E t = t .
The minimum of
[(
) 2 ]
E T+1 − X 1 X 2 X T
over all the measurable functions T → is obtained for
⋆ X 1 X 2 X T = EX T+1 X 1 X 2 X T
= E T+1 X 1 X 2 X T
Proof
An easy way to get the announced result consists in noting that
[(
) 2 ]
E T+1 − X 1 X 2 X T
[(
= E T+1 − ⋆ X 1 X 2 X T
) 2 ]
+ ⋆ X 1 X 2 X T − X 1 X 2 X T
[(
) 2 ]
= E T+1 − ⋆ X 1 X 2 X T
[(
) 2 ]
+ E ⋆ X 1 X 2 X T − X 1 X 2 X T
which is clearly minimal for ≡ ⋆ .