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

140 Actuarial Modelling of Claim Counts
mixing distributions examined in Chapter 2 are all unimodal. Henceforth, we denote by
s 0 b m − unim the unimodal moment space of all the random variables of this type.
In the following, we use the notation nim z, with m, z ∈ , for the Uniform
distribution on the interval minm z maxm z: ifmz, it is the law
with constant probability density function equal to 1/m − z on z m; ifm = z, itisthe
law degenerated at the point m. Given some random variable Z, we denote by nim Z
the mixed Uniform distribution with random extremal point Z as mixing parameter.
A convenient representation of unimodal distributions is provided by Khinchine’s theorem
(see, e.g., Theorem 1.3 in Dharmadhikari & Joag-Dev (1988)). This theorem states that a
random variable Y has a unimodal law with a mode at 0 if, and only if, Y is distributed as
UZ where U and Z are two independent random variables and U ∼ ni0 1. This condition
can be rewritten as Y ∼ ni0Z for some random variable Z.
Now, let X be any random variable valued in 0b and with a unique mode at m. By
Khinchine’s theorem,
X ∼ nim˜Z where ˜Z = m + Z (3.10)
Note that Z is valued in −m b − m. Moreover, the moments j of X and ˜ j of ˜Z are
linked by simple relations. Indeed, we have
∫ ( 1
∫ m
)
j =
x j dx dF˜Z
m − z
z
x=z
∫
m j+1 − z j+1
=
dF˜Z z
m − zj + 1
= 1 ∫
m j + m j−1 z + m j−2 z 2 +···+z j dF˜Z
j + 1
z
= 1
j + 1 mj + m j−1˜ 1 + m j−2˜ 2 +···+˜ j j = 1 2 (3.11)
From (3.11), we get
˜ j = j + 1 j − mj j−1 j = 1 2 (3.12)
Let us now come back to (3.8) in s 0 b m − unim . Specifically, we would like to
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 m − unim (3.13)
As shown in Denuit, De Vylder & Lefèvre (1999), the random variables X s⋆
min
and Xs⋆ max
involved in (3.13) give bounds on EX for every s-convex function . Finding the
solution of (3.13) for X in s 0 b m − unim thus amounts to solving the corresponding
problem in s 0 b ˜.
Tables 3.8–3.9 give explicit expressions for the improved extremal distributions, also for
values of s up to five. In these tables ∑ k
i=1 p ini i i ,0≤ p i ≤ 1, i i ∈ , i = 1 2k,
represents a mixture of the distributions ni i i , with respective weights p i .