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

224 Actuarial Modelling of Claim Counts
We only give hereafter a short non technical description of the fundaments of Extreme
Value Theory; for more details, we refer the reader to Beirlant ET AL. (2004). Considering
a sequence of independent and identically distributed random variables (claim severities,
say) X 1 X 2 X 3 , most classical results from probability and statistics that are relevant
for insurance are based on sums S n = ∑ n
i=1 X i. Let us mention the Law of Large Numbers
and the Central Limit Theorem, for instance. Another interesting yet less standard statistic
for the actuary is M n = maxX 1 X n the maximum of the n claims. Extreme Value
Theory mainly addresses the following question: how does M n behave in large samples (i.e.
when n tends to infinity)? Of course, without further restriction, M n obviously diverges to
+. Once M n is appropriately centered and normalized, however, it may converge to some
specific limit distribution (of three different types, according to the fatness of the tails of the
X i s). In insurance applications, heavy tailed distributions are most often encountered. Such
distributions have survival functions that decay like a power function (in contrast to the
Gamma, Inverse Gaussian or LogNormal survival functions, for instance, which all decay
exponentially to zero). A prominent example of a heavy tailed distribution is the Pareto
distribution, widely used by actuaries.
Excess Over Threshold Approach and Generalized Pareto Distribution
The traditional approach to Extreme Value Theory is based on extreme value limit
distributions. Here, a model for extreme losses is based on the possible parametric form of
the limit distribution of maxima. A more flexible model is known as the ‘Excesses Over
Threshold’ method. This approach appears as an alternative to maxima analysis for studying
the extreme behaviour of some random variables. Basically, given a series X 1 X n of
independent and identically distributed random variables, the ‘Excesses Over Threshold’
method analyses the series X i − uX i >u, i = 1n, of the exceedances of the variable
over a high threshold u. Mathematical theory supports the Poisson distribution for the number
of exceedances combined with independent excesses over the threshold.
Let F u stand for the common cumulative distribution function of the X i − uX i >us;
F u thus represents the conditional distribution of the losses, given that they exceed the
threshold u. The two-parameter Generalized Pareto distribution function G · provides a
good approximation to the excess distribution F u over large thresholds. This two-parameter
family is defined as
G x = G
( x
)
>0
where
{
1 − 1 + x −1/ if ≠ 0
G x =
1 − exp−x if = 0
with x ≥ 0if ≥ 0 and x ∈ 0 −1/ if 0, the type II Pareto distribution when