3. Assessing the Adequacy: Mean Excess Functions
- r =
empme
(x, t)
-
returns the value of the empirical mean excess function
based on the real vector x at all elements of the vector t
- r =
gpme
(gamma, t)
-
returns the value of the GP mean excess function of a GP
distribution with the shape parameter gamma at
all elements of the vector t
- r =
gp1me
(alpha, t)
-
returns the value of the mean excess function of a Pareto (GP1)
distribution with shape parameter alpha at
all elements of the vector t
|
Let X be a random variable with df F. Then, the mean excess
function of F is
eF(t) := E(X-t|X>t).
If F includes location and scale parameters
and
,
then
The mean excess function of a Pareto (GP1) distribution
is
For the generalized Pareto distribution
the mean excess
function is given by
for t>0, if
and
,
if
Notice that the mean excess function does not exist for
(
in the Pareto (GP1) model).
Mean excess functions are linear if, and only if, F is a generalized
Pareto distribution. Therefore, the empirical
mean excess function
can be employed to check
if a GP modeling of a given data set is plausible.
Moreover, by comparing the empirical mean excess function and a parametric
one, fitted by an estimator, one obtains a visual tool to control
the result of the estimation. A stronger deviation from the empirical
mean excess function shows that an estimator may not be
applicable. In the example provided in Section 8, we apply this tool
to make a choice between two different parametric estimation procedures.