13.2 Generalized Pareto Distributions


r = 31276 randx ("gp", n, gamma)
returns a vector with n pseudorandom GP variables under the shape parameter gamma
r = 31279 pdfx ("gp", x, gamma)
returns the value of the GP density with shape parameter gamma for all elements of the vector x
r = 31282 cdfx ("gp", x, gamma)
returns the value of the GP distribution function with shape parameter gamma for all elements of the vector x
r = 31285 qfx ("gp", x, gamma)
returns the value of the GP quantile function with shape parameter gamma for all elements of the vector x

In this section, we introduce a parametric model for those data $ y_1,\dots,y_k$ which exceed a threshold $ t$, where $ x_1, \dots,x_n$ are the original data. Likewise, one may consider the $ k$ largest values $ (y_1, \dots, y_k) = (x_{n-k+1:n},\dots,x_{n:n})$ of the data $ x_i$. Generalized Pareto (GP) distributions constitute adequate models for such data (Reiss and Thomas; 1997). They consist of the following three submodels:

(i) Exponential (GP0) $ W_0(x) = 1-e^{-x}$, $ x \ge 0,$
(ii) Pareto (GP1) $ W_{1,\alpha}(x) = 1-x^{-\alpha}$, $ x \ge 1, \, \alpha > 0$,
(iii Beta (GP2) $ W_{2,\alpha}(x) = 1-(-x)^{-\alpha}$ , $ -1 \le x \le 0, \, \alpha < 0$.

Again, one can unify these distributions by using the parameterization with $ \gamma=1/\alpha$. We have

$\displaystyle W_\gamma (x) = \left\{
\ \begin{array}{ll}
\ 1-(1+\gamma x)^{-1/\...
...\, \gamma < 0 \\
\ 1-e^{-x} & x \ge 0, \, \gamma = 0
\ \end{array}\ \right..
$

A mathematical justification of the modeling is obtained by a limit theorem. Assume that $ X$ is a random variable with df $ F \in {\cal D}(G_\gamma)$, and consider the conditional distribution

$\displaystyle F^{[t]}(x) := P(X \le x \vert X > t),\quad x>t,
$

which is the common df of the exceedances $ y_j$ above the threshold $ t$. For the sequence of thresholds $ t_n = a_n t + b_n$ the convergence

$\displaystyle F^{[a_n t + b_n]}(a_n (t+s) + b_n) \to 1 + \log G_\gamma(s/(1+\gamma t))
$

holds (Falk, Hüsler and Reiss; 1994). Formally, the relation $ W_\gamma=1+\log G_\gamma$ holds between EV and GP dfs $ G_\gamma$ and $ W_\gamma$.

GP distributions are the only continuous dfs such that

$\displaystyle F^{[t]} (a_t x + b_t) = F(x).
$

More precisely, the relation

$\displaystyle W_{\gamma,\mu,\sigma}^{[t]} = W_{\gamma,t,\sigma + \gamma(t-\mu)}
$

holds. The location parameter of the truncated df is the truncation point $ t$, while the shape parameter $ \gamma$ remains unchanged.

The quantlets concerning densities, distribution and quantile functions of generalized Pareto distributions as well as the generation of pseudorandom variables are shown at the beginning of this section. Again, the routines belonging to the von Mises parameterization are merely displayed. One can address the three submodels by providing the names "gp0", "gp1" and "gp2" instead of "gp". Within the GP0 model, the shape parameter is not required.