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
):
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image004.png)
If
then
its domain is
.
The mean excess function of a probability distribution is
defined as:
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image007.png)
If
then
then mean excess function for this distribution is as follows (for
):
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image010.png)
Let
so
and
.
Let
.
Then:
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image015.png)
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image016.png)
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image017.png)
![](I/ERMMTGPDMeanExcessLinearInExceedanceThreshold_files/image018.png)
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).