= Generalized Pareto distribution
{title2=$G_{\xi,\sigma}(y)$}
{wiki}
For scale $\sigma>0$ and shape $\xi$, the generalized Pareto distribution on nonnegative excesses has <cumulative distribution function>
$$
G_{\xi,\sigma}(y)=1-(1+\xi y/\sigma)^{-1/\xi},\qquad y\geq0,\quad1+\xi y/\sigma>0.
$$
The case $\xi=0$ is the continuous limit $1-e^{-y/\sigma}$. Positive $\xi$ gives a power-law tail, zero gives an <exponential distribution>, and negative $\xi$ gives a finite upper endpoint $-\sigma/\xi$. Its role in modelling high-threshold excesses is explained by the <Pickands-Balkema-de Haan theorem>.
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