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

138 Actuarial Modelling of Claim Counts
Table 3.6
s = 1to5.
Probability distribution of X s
max ∈ s 0 b achieving the upper bound in (3.8) for
s Support points Probability masses
1 b 1
b −
2 0
1
b
b
1
b
b
3 1 − 2
b − 1 2
b − 1 b − 1 2 + 2 − 2 1
b
2 − 2 1
b − 1 2 + 2 − 2 1
4 0 1− p 1 − p 2
3 − b 2
p
2 − b 1 =
2 − b 1 3
1 3 − b 2 3 − 2b 2 + b 2 1
b p 2 =
5 z + = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 + √
2 1 − b 3 − b 2 − 2 − b 1 2
z − = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 − √
2 1 − b 3 − b 2 − 2 − b 1 2
b
1 3 − 2 2
b 3 − 2b 2 + b 2 1
p + = 2 − b + z − 1 + bz −
z + − z − z + − b
p − = 2 − b + z + 1 + bz +
z − − z + z − − b
1 − p + − p −
Where = 1 − b 4 − b 3 − 2 − b 1 3 − b 2 2 − 4 1 − b 3 − b 2 − 2 −
b 1 2 2 − b 1 4 − b 3 − 3 − b 2 2
Approximations Based on First Moments
A given random variable X (think of the risk parameter i in credibility applications) with
known moments k , k = 1 2, can be approximated either by X s
min
or Xs max involved in
(3.8). An alternative approximation mixing these two extremal variables is also available.
Let us denote as
and
s
=
s =
min
X∈ s 0b
max
X∈ s 0b
EX s = E [ X s
min s]
EX s = E [ X s
max s]
the lower and upper bounds involved in (3.8). Explicit expressions of s
and of s for s up
to five are listed in Table 3.7, in the notation of Tables 3.5–3.6. The quantity s − s
can
be considered as the ‘width’ of s 0 b as explained in Denuit (2002).