13.3 Assessing the Adequacy: Mean Excess Functions


r = 31452 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 = 31455 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 = 31458 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

$\displaystyle e_F(t) := E(X-t\vert X>t).
$

If $ F$ includes location and scale parameters $ \mu$ and $ \sigma$, then

$\displaystyle e_{F_{\mu,\sigma}} = \sigma e_F \left({t-\mu \over \sigma}\right).
$

The mean excess function of a Pareto (GP1) distribution $ W_{1,\alpha}$ is

$\displaystyle e_{W_{1,\alpha}}(t) = {t\over\alpha-1}, \quad t>1, \alpha>1.
$

For the generalized Pareto distribution $ W_\gamma,$ the mean excess function is given by

$\displaystyle e_{W_\gamma}(t)= {1+\gamma t \over 1 - \gamma}
$

for $ t>0$, if $ 0 \le \gamma < 1,$ and $ 0 < t < -1/\gamma$, if $ \gamma < 0.$ Notice that the mean excess function does not exist for $ \gamma\ge 1$ ( $ \alpha\le 1$ 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

$\displaystyle e_n(t) = { \sum_{i=1}^n (x_i - t) I(t < x_i) \over
\ \sum_{i=1}^n I (t < x_i) },
\quad x_{1:n} \le t < x_{n:n},
$

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 13.8, we apply this tool to make a choice between two different parametric estimation procedures.