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

Credibility Models for Claim Counts 137
determination of conditions on 0 b 1 2 s−1 so that s 0 b is not void, or
the obtention of the elements in s 0 b with the minimal number of support points.
These elements possess some extremal properties and will be used here in connection with
credibility formulas. Henceforth, we tacitly assume that 0b and 1 2 s−1 are such
that the associated moment space s 0 b is not void and is not a singleton (i.e.,
s 0 b consists of at least two distinct distribution functions).
De Vylder’s Moment Problem
De Vylder (1996, Section 8.3) investigated the following problem: within s 0 b ,
determine the random variables X s
min
and Xs max such that the inequalities
EX s
min s ≤ EX s ≤ EX s
max s hold for all X ∈ s 0 b (3.8)
Explicit solutions to (3.8) are available for s up to five. The supports of the extremal
distributions are given in Tables 3.5–3.6.
As shown in Denuit, De Vylder & Lefèvre (1999), the random variables X s
X s
min and
max involved in (3.8) give bounds on EX for every function that is s − 2 times
differentiable, with a convex s − 2th derivative (such functions are called s-convex; see
Roberts & Varberg (1973) for details). In that context, X s
min
is called the s-convex
minimum, and Xmax s is called the s-convex maximum.
Table 3.5
s = 1to5.
Probability distribution of X s
min ∈ s0 b achieving the lower bound in (3.8) for
s Support points Probability masses
1 0 1
2 1 1
3 0
2 − 2 1
2
2
1
2 1
2
4 r + = 3 − 1 2 + √ 3 − 1 2 2 − 4 2 − 2 1 1 3 − 2 2
2 2 − 2 1
r − = 3 − 1 2 − √ 3 − 1 2 2 − 4 2 − 2 1 1 3 − 2 2
2 2 − 2 1
1 − r −
r + − r −
1 − 1 − r −
r + − r −
5 0 1− q + − q −
t + = 1 4 − 2 3 + √ 1 4 − 2 3 2 − 4 1 3 − 2 2 2 4 − 2 3
2 1 3 − 2 2 q + = 2 − t − 1
t + t + − t −
t − = 1 4 − 2 3 − √ 1 4 − 2 3 2 − 4 1 3 − 2 2 2 4 − 2 3
2 1 3 − 2 2 q − = 2 − t + 1
t − t − − t +