Showing that the Mean Excess Function of
a Generalised
Pareto Distribution is linear in the exceedance threshold (for a specific
range of values of the distribution’s shape parameter)
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If a random variable, ,
is distributed according to a generalised Pareto distribution, ,
then it has the following probability density function (for ):
If then
its domain is .
The mean excess function of a probability distribution is
defined as:
If then
then mean excess function for this distribution is as follows (for ):
Let so
and
.
Let .
Then:
This is linear in as
desired. A consequence is that we can test visually whether a data set appears
to be coming from a GPD by plotting the empirical mean excess function and
seeing if it appears to be linear (and we can also estimate from
its slope if it is linear).