The GPD fitting procedure models the magnitude of losses above a threshold but ignores when those losses occur. The point process framework unifies both: it models threshold exceedances as events in time and their sizes simultaneously, embedding the GEV and GPD models into a single object — the two-dimensional Poisson point process (PPP). This is the theoretically cleanest formulation of the peaks-over-threshold approach. The PPP likelihood factorises into the GPD likelihood (for excess sizes) and a Poisson likelihood (for exceedance timing), showing that these two aspects are statistically separable. It also exposes the right way to handle the complication of clustering in financial time series: either decluster the data to approximate iid conditions, or model the cluster-generating mechanism dynamically.
Learning objectives. By the end of this note you should be able to:
- state the definition of a Poisson point process and its intensity measure;
- derive the Poisson limit for the number of threshold exceedances;
- write the two-dimensional intensity function of the POT model and its implied marginals (GPD for excesses, GEV for maxima);
- write the POT likelihood and explain why it factorises into GPD and Poisson components;
- state the problem with applying the standard POT model to GARCH-driven financial series and describe the declustering remedy.
In this lesson
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- Point processes and the Poisson limit for exceedances
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- The two-dimensional POT model
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- Statistical estimation and likelihood factorisation
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- Financial data and the clustering problem
Before you start
- The Generalised Pareto Distribution and Peaks-Over-Threshold
- The Generalised Extreme Value Distribution and Block Maxima
The full lesson, with worked examples and exercise solutions, is for members.
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