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

228 Actuarial Modelling of Claim Counts
of and coming from the linear fit to the empirical mean excess function could also be
used, as shown below.
Moment Method
The mean and the variance of the Generalized Pareto distribution are respectively given
by /1 − provided u
− u = ∑ n
i=1 x i − uIx i >u
#x i x i >u
where IA = 1 if the event A did occur and 0 otherwise. This means that eu is estimated by
the sum of exceedances over the threshold u divided by the number of data points exceeding
the threshold u.
Usually, the mean excess function is evaluated on the observations of the sample. Denoting
the sample observations arranged in ascending order as x 1 ≤ x 2 ≤···≤x n , we have in
this case
ê n x k = 1 n−k
∑
x
n − k k+j − x k
It is easily checked that when X has a Generalized Pareto distribution function G , the
mean excess function is a linear function in u
eu =
j=1
1 − +
1 − u
provided +u > 0. Hence, the idea is to determine, on the basis of the graph of the empirical
estimator of the excess function ê n , a region u + where ê n t becomes approximately
linear for t ≥ u. The intercept and slope of a straight line fit to ê n determine the estimations
of and .