Modelling interexceedance times of temporally clustered extreme events
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Date
2025
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Abstract
This dissertation presents a unified framework for modelling interexeedance times of temporally clustered extreme events. Building on two established models, we introduce a generalised model, which combines underlying stationary magnitudes with random, heavy-tailed waiting times. When a sufficiently high threshold is used to define exceedances, the resulting interexceedance times follow a mixture of a Dirac measure at zero and a Mittag-Leffler distribution.
The generalized mixed distribution is characterised by three parameters: the tail index, the extremal index, and a scale parameter. To estimate them from data, we propose two minimum distance estimators (CMmod1 and CMmod2) based on modified Cramér-von Mises distances. Their weak consistency is shown theoretically, and their finite sample performance is assessed through extensive simulations. Results highlight CMmod2's robustness to threshold selection and parameter constellations. The estimators are also evaluated against existing methods in the special cases of the compound Poisson process and the fractional Poisson process. Despite requiring the estimation of an additional parameter, CMmod1 and CMmod2 show competitive accuracy, suggesting that explicit model selection may be avoidable.
A case study on North Atlantic cyclone data illustrates the practical value of the FCPP, revealing spatial differences in clustering behaviour and confirming the real-world applicability of the proposed methodology.
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Extremwertstatistik, Markierte Punktprozesse, Minimum Distanz Schätzer, Mischverteilung
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Extremwertstatistik, Markierter Punktprozess, Schätzfunktion, Zusammengesetzte Verteilung