13.1 Extreme Value Models


r = 31051 randx ("ev", n, gamma)
returns a vector with n pseudorandom EV variables under the shape parameter gamma
r = 31054 pdfx ("ev", x, gamma)
returns the values of the EV density with shape parameter gamma for all elements of the vector x
r = 31057 cdfx ("ev", x, gamma)
returns the values of the EV distribution function with shape parameter gamma for all elements of the vector x
r = 31060 qfx ("ev", x, gamma)
returns the value of the EV quantile function with shape parameter gamma for all elements of the vector x

In this section, we introduce a parametric model for maxima $ y_i = \max\{x_{i,1}, \dots,\linebreak[3]x_{i,n}\}$, $ i=1,\dots,m,$ where the $ x_{i,j}$ may not be observable. Such data occur, e.g. when annual flood maxima or monthly maxima of temperatures are recorded. In the first case, we have $ n=365$ while $ m$ is the number of observed years. We mention a limit theorem which suggests a parametric modeling of such maxima using extreme value (EV) distributions.

Assume that $ X_1, \dots, X_n$ are independent random variables with common distribution function (df) F, and let $ Y:=\max\{X_1, \dots, X_n\}.$ Then, $ P\{Y \le t\} = F^n(t)$ is the df of $ Y.$ If, for an appropriate choice of constants $ a_n$ and $ b_n$, the dfs of the maxima converge to a continuous limiting distribution function, i.e.

$\displaystyle F^n(a_n x + b_n) \to G(x), \quad n \to \infty,
$

then $ G$ is equal to one of the following types of distributions (a result due to Fisher and Tippett; 1928):
(i) Gumbel (EV0) $ G_0(x) = \exp (-e^{-x})$, $ x \in \mathbb{R},$
(ii) Fréchet (EV1) $ G_{1,\alpha}(x) = \exp(-x^{-\alpha})$, $ x \ge 0, \, \alpha > 0$,
(iii) Weibull (EV2) $ G_{2,\alpha}(x) = \exp (-(-x)^{-\alpha})$, $ x \le 0, \, \alpha < 0$.
Moreover, the df $ F$ is said to belong to the domain of attraction of the EV distribution $ G$ (in short, $ F \in {\cal D}(G)$). These EV distributions are also the limiting distributions of maxima of dependent random variables under weak conditions (see Leadbetter and Nandagopalan; 1989).

By employing the reparametrization $ \gamma=1/\alpha$ and appropriate scale and location parameters, one can unify these models in the von Mises parameterization by

$\displaystyle G_\gamma(x) = \left\{
\ \begin{array}{ll}
\ \exp(-(1+\gamma x)^{-...
...
\ \exp(-e^{-x}),
\ & x \in \mathbb{R}, \, \gamma = 0.
\ \end{array}\ \right.
$

Notice that $ G_\gamma(x) \to G_0(x)$ for $ \gamma\to 0.$ Moreover, the relation $ G_\gamma = G_{i,\alpha,-\alpha,\vert\alpha\vert}$ holds with $ \alpha=1/\gamma$, $ i=1$ for $ \gamma>0$, and $ i=2$ for $ \gamma<0$.

The quantlets concerning densities, distribution and quantile functions (qfs) of extreme value distributions as well as the generation of pseudorandom variables are listed at the beginning of this section. The routines belonging to the von Mises parameterization are merely displayed. One can address the three submodels by providing the names "ev0", "ev1" and "ev2" in place of "ev". Notice the shape parameter is not required within the EV0 model. For example, use

  pdfx("ev2", x, alpha)
to calculate the Weibull density with shape parameter $ \alpha$. In addition, location and scale parameters $ \mu$ and $ \sigma$ can easily be included, e.g. take
  r = mu + sigma * randx("ev", n, gamma)
to generate a data set with $ n$ independent realizations under $ G_{\gamma,\mu,\sigma}$.