Generalized Pareto distribution ($G_{\xi ,\sigma }(y)$)

For scale $\sigma >0$ and shape $\xi$, the generalized Pareto distribution on nonnegative excesses has cumulative distribution functionThe 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.

The peaks-over-threshold method fits a generalized Pareto distribution to exceedances of a high threshold and also estimates their frequency. Together these determine tail probabilities and return levels. For dependent observations, threshold exceedances may occur in clusters; declustering of extremes can separate extreme events before fitting a model for independent events.

Declustering groups dependent threshold exceedances into extreme-event clusters and retains a representative, commonly the cluster maximum. It aims to make the events used in a peaks-over-threshold method approximately independent. The event rate must be estimated consistently with the retained clusters when computing return levels.

For a distribution in an extreme-value domain of attraction, the conditional distribution of the excess above a high threshold is asymptotically approximated by a generalized Pareto distribution with a threshold-dependent scale and a limiting shape parameter. This theorem justifies the peaks-over-threshold method. Threshold selection must balance approximation error against the reduced number of excess observations.