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

156 Actuarial Modelling of Claim Counts
In the model A1–A2 of Definition 3.1, we intuitively feel that the following statements
are true:
Statement S1
Statement S2
Statement S3
i ‘increases’ in the past claims N i•
N iTi +1 ‘increases’ in the past claims N i•
N iTi +1 and N i• are ‘positively dependent’.
This section aims to precisely define the meaning of ‘increases’ in statements S1 and S2, as
well as the nature of the ‘positive dependence’ involved in statement S3. The proofs will be
omitted because of their technical nature; for a detailed study, we refer the reader to Denuit
ET AL. (2005, Chapter 7). Note that we present the results in terms of stochastic dominance
whereas in fact the stronger (but less intuitive) likelihood ratio order applies.
3.5.2 Stochastic Order Relations
In order to formalize the increasingness involved in statements S1–S2, our study will
extensively resort to stochastic orderings. Therefore, we recall in this section the definition of
stochastic dominance, as well as some intuitive intepretations. Given two random variables
X and Y , X is said to be smaller than Y in the stochastic dominance, written as X ≼ ST Y ,if
PrX>t≤ PrY>tfor all t ∈
We see that a ranking in the ≼ ST -sense translates the intuitive meaning of ‘being smaller
than’ in probability models: indeed, we compare the probability that both random variables
exceed some given threshold t, and the smallest one in the ≼ ST -sense has the smallest
probability of exceeding the threshold. If M and N are two counting random variables then
M ≼ ST N ⇔
+∑
j=k
PrM = j ≤
+∑
j=k
PrN = j for all k = 0 1
One intuitively feels that a random variable N following the Poisson distribution with
mean gets bigger as increases. The next implication formalizes this intuitive statement:
≤ ′ ⇒ N ≼ ST N ′
The oi family thus increases in its parameter in the ≼ ST -sense.
3.5.3 Comparisons of Predictive Distributions
We then have the following results that formalize statements S1 and S2. First, i increases
in the past claims N i• in the ≼ ST -sense, that is
i N i• = k ≼ ST i N i• = k ′ for k ≤ k ′ (3.18)
⇔ Pr i >tN i• = k ≤ Pr i >tN i• = k ′ whatever t, provided k ≤ k ′