User:Isheden/sandbox3

Generalized Pareto distribution
A standard generalized Pareto distribution has a cumulative distribution function
 * $$G_{\xi}(x) = \begin{cases}

1 - (1+ \xi x)^{-1/\xi} & \text{for }\xi \neq 0, \\ 1 - e^{-x} & \text{for }\xi = 0, \end{cases} $$ where $$ x \geq 0 $$ when $$ \xi \geq 0 \,$$, and $$ 0 \leq x \leq 1 /\xi $$ when $$ \xi < 0 \,$$.

The related location-scale family $$G_{(\xi,\mu,\sigma)}$$ (3-parameter GPD) is
 * $$G_{(\xi,\mu,\sigma)}(x) = \begin{cases}

1 - \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi} & \text{for }\xi \neq 0, \\ 1 - \exp \left(-\frac{x-\mu}{\sigma}\right) & \text{for }\xi = 0, \end{cases} $$

where $$ x \geqslant \mu $$ when $$ \xi \geqslant 0 \,$$, and $$ \mu \leqslant x \leqslant \mu - \sigma /\xi $$ when $$ \xi < 0 \,$$.

Alternative $$G_{(\xi,\beta)}$$
 * $$G_{(\xi,\beta)}(x) = \begin{cases}

1 - \left(1+ \xi \frac{x}{\beta}\right)^{-1/\xi} & \text{for }\xi \neq 0, \\ 1 - \exp \left(-\frac{x}{\beta}\right) & \text{for }\xi = 0, \end{cases} $$

where $$ x \geqslant 0 $$ when $$ \xi \geqslant 0 \,$$, and $$ 0 \leqslant x \leqslant - (\beta /\xi) $$ when $$ \xi < 0 \,$$.

Pickands?