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

226 Actuarial Modelling of Claim Counts
expected that the behaviour of the differences corresponding to the extreme observations
will be different from the one corresponding to the non-extreme observations. This change
of behaviour will appear as a change point of the series of differences.
To identify the change point in a series, a sequential version of the Mann-Kendall test is
applied. In this test, the normalized series U i is defined as
where U ∗
i
U i = U ∗
√
i
− ii−1
4
ii−12i+5
72
= ∑ i
k=2 n k, and n k is the number of values in 2 k lesser than k . Another
series, denoted by U p , is calculated applying the same procedure to the series of the
differences from the end to the start, n 2 , instead of from the start to the end. The
intersection point between these two series, U i and U p , determines a probable change point
that will be significant if it exceeds a high Normal percentile.
Since, usually, these techniques can only provide approximative information about the
threshold, simultaneous application of them is highly recommended in order to get more
reliable results.
Application to Claim Costs Recorded in Portfolio C
The Generalized Pareto index plot is shown in Figure 5.1. We see that the estimates of
tail parameter roughly stabilize after E85000. The Gertensgarbe plot gives a threshold of
E104 397 (which corresponds to the 17th largest loss). The p-value of the Mann-Kendall
0.4
0.6
Estimation of ξ
0.8
1.0
50000 100000 150000
Threshold
Figure 5.1 Generalized Pareto index plot for the claim costs in Portfolio C.