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

22 Actuarial Modelling of Claim Counts
The interpretation of the probability density function is that
Prx ≤ X ≤ x + h ≈ fxh for small h>0
That is, the probability that a random variable, with an absolutely continuous probability
distribution, takes a value in a small interval of length h is given by the probability density
function times the length of the interval.
A general type of distribution function is a combination of the discrete and (absolutely)
continuous cases, being continuous apart from a countable set of exception points
x 1 x 2 x 3 with positive probabilities of occurrence, causing jumps in the distribution
function at these points. Such a distribution function F X can be represented as
F X x = 1 − pF c
X x + pF d
X x x ∈ (1.22)
for some p ∈ 0 1, where F c
X is a continuous distribution function and F d
X
distribution function with support d 1 d 2 .
Let us assume that F X is of the form (1.22) with
pF d
X t = ∑ (
)
F X d n − F X d n − = ∑ PrX = d n
d n ≤t
d n ≤t
is a discrete
where d 1 d 2 denotes the set of discontinuity points and
Then,
∫ t
1 − pF c
X t = F X t − pF d
X t = f c
X xdx
−
EX = ∑ )
d n
(F X d n − F X d n − +
n≥1
∫ +
−
xf c
X xdx (1.23)
=
∫ +
−
xdF X x
where the differential of F X , denoted as dF X , is defined as
{
FX d n − F X d n − if x = d n
dF X x =
f c
X xdx otherwise
This unified notation allows us to avoid tedious repetitions of statements like ‘the proof
is given for continuous random variables; the discrete case is similar’. A very readable
introduction to differentials and Riemann–Stieltjes integrals can be found in Carter & Van
Brunt (2000).
1.4.2 Heterogeneity and Mixture Models
Definition
Mixture models are a discrete or continuous weighted combination of distributions aimed
at representing a heterogeneous population comprised of several (two or more) distinct subpopulations.
Such models are typically used when a heterogeneous population of sampling