Showing that the Mean Excess Function of
Pareto Distribution is linear in the exceedance threshold (for a specific
range of values of the distribution’s shape parameter)
[this page | pdf | back links]
If a random variable, ,
is distributed according to a generalised Pareto distribution, ,
then it has the following probability density function (for ):
its domain is .
The mean excess function of a probability distribution is
then mean excess function for this distribution is as follows (for ):
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).