2. Generalized Pareto Distributions


r = 1042 randx ("gp", n, gamma)
returns a vector with n pseudorandom GP variables under the shape parameter gamma
r = 1045 pdfx ("gp", x, gamma)
returns the value of the GP density with shape parameter gamma for all elements of the vector x
r = 1048 cdfx ("gp", x, gamma)
returns the value of the GP distribution function with shape parameter gamma for all elements of the vector x
r = 1051 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 xi. Generalized Pareto (GP) distributions constitute adequate models for such data (Reiss and Thomas; 1997). They consist of the following three submodels:

(i) Exponential (GP0) W0(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

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

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

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

which is the common df of the exceedances yj above the threshold t. For the sequence of thresholds tn = an t + bn the convergence

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

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

F[t] (at x + bt) = F(x).

More precisely, the relation

\begin{displaymath}
W_{\gamma,\mu,\sigma}^{[t]} = W_{\gamma,t,\sigma + \gamma(t-\mu)}
\end{displaymath}

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.



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