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

30 Actuarial Modelling of Claim Counts
to experience another accident. In a longitudinal setting, actual and future outcomes are
directly influenced by past values, and this causes a substantial change over time in the
corresponding distribution.
Since with event count data we only observe the total number of events at the end
of the period, contagion, like heterogeneity, is an unobserved, within-observation process.
For research problems where both heterogeneity and contagion are plausible, the different
underlying processes are not distinguishable with aggregate count data because they both
lead to the same probability distribution for the counts. One can still use this distribution to
derive fully efficient and consistent estimates, but this analysis will only be suggestive of
the underlying process.
Poisson Limiting Form
The Negative Binomial distribution has a Poisson limiting form if V = 1 → 0. This result
a
can be recovered from the sequence of the probability generating functions, noting that
(
lim
a↗
a
a − d1 − z
) a (
= lim 1 − d ) −a
a↗ a 1 − z = exp−d1 − z
that is seen to converge to the probability generating function of the Poisson distribution
with parameter d.
Derivation as a Compound Poisson Distribution
A different type of heterogeneity occurs when there is clustering. If it is assumed that
the number of clusters is Poisson distributed, but the number of individuals in a cluster is
distributed according to the Logarithmic distribution, then the overall distribution is Negative
Binomial. In an actuarial context, this amounts to recognizing that several vehicles can
be involved in the same accident, each of the insured drivers filing a claim. Therefore, a
single accident may generate several claims. If the number of claims per accident follows
a Logarithmic distribution, and the number of accidents over the time interval of interest
follows a Poisson distribution, then the total number of claims for the time interval can be
modelled with the Negative Binomial distribution.
Let us formally establish this result. Recall that the random variable M has a Logarithmic
distribution if
PrM = k =
k
−k ln1 −
k= 1 2
where 0