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

Credibility Models for Claim Counts 157
This relation is transmitted to the number of claims, in the sense that
N iTi +1N i• = k ≼ ST N iTi +1N i• = k ′ for k ≤ k ′ (3.19)
⇔ PrN iTi +1 >jN i• = k ≤ N iTi +1 >jN i• = k ′ whatever j, provided k ≤ k ′
3.5.4 Positive Dependence Notions
In order to formalize the positive dependence involved in statement S3, we will present
some concepts of dependence related to ≼ ST . The study of concepts of positive dependence
for random variables, started in the late 1960s, has yielded numerous useful results in both
statistical theory and applications. Applications of these concepts in actuarial science recently
received increased interest.
Let us formalize the positive dependence existing between the two components of a
random couple (i.e. the fact that large values of one component tend to be associated with
large values for the other). Formally, let X = X 1 X 2 be a bivariate random vector. Then,
X is positive regression dependent (PRD, for short) if X 2 X 1 = x 1 ≼ ST X 2 X 1 = x
1 ′ for
all x 1 ≤ x
1
′ and X 1X 2 = x 2 ≼ ST X 1 X 2 = x
2 ′ for all x 2 ≤ x
2 ′ . PRD imposes stochastic
increasingness of one component of the random couple in the value assumed by the other
component in the ≼ ST -sense. This dependence notion is thus rather intuitive.
PRD naturally extends to higher dimension. Specifically, let X = X 1 X n be a
n-dimensional random vector. Then,
(i) X is conditionally increasing (CI, for short) if
X i X j = x j j ∈ J ≼ ST X i X j = x ′ j j ∈ J
whenever x j ≤ x
j ′ , j ∈ J, J ⊂ 1 2nand i ∉ J.
(ii) X is conditionally increasing in sequence (CIS, for short) if X i is stochastically increasing
in X 1 X i−1 , for i ∈ 2ni.e.
X i X 1 = x 1 X i−1 = x i−1 ≼ ST X i X 1 = x ′ 1 X i−1 = x ′ i−1
whenever x j ≤ x
j ′ j∈ 1i− 1.
The conditional increasingness in sequence is interesting when there is a natural order in the
components of X, induced by obervation times for instance.
3.5.5 Dependence Between Annual Claim Numbers
The total claim number N i• reported in the past periods and the claim number N iTi +1 for the
next coverage period are PRD. The fact that N i• and N iTi +1 are PRD completes the statement
(3.19). This provides a host of useful inequalities. In particular, whatever the distribution of
i , the credibility coefficient E i N i• = k is increasing in k, which is easily deduced from
(3.18).
Considering the dependence existing between the components of N i , i.e. between the N it s,
t = 1 2T i , we can prove that N i is CI.