3. Assessing the Adequacy: Mean Excess Functions


r = 1076 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 = 1079 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 = 1082 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 $\mu$ and $\sigma$, then

\begin{displaymath}

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

\end{displaymath}

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

\begin{displaymath}

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

\end{displaymath}

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

\begin{displaymath}

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

\end{displaymath}

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

\begin{displaymath}

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},

\end{displaymath}

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.



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