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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).

 

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