Representations of max-stable processes based on single extreme events

TL;DR

This paper develops new inference methods for max-stable processes using single extreme events, leveraging simpler incremental representations and a peaks-over-threshold approach to utilize more data.

Contribution

It introduces a novel inference framework based on incremental representations and single extreme events, connecting max-stable and M3 process representations.

Findings

Convergence results for process increments are established.

Shape functions of M3 processes can be derived from single extreme events.

Transformation formulas link incremental and M3 representations.

Abstract

This paper provides the basis for new methods of inference for max-stable processes \xi\ on general spaces that admit a certain incremental representation, which, in important cases, has a much simpler structure than the max-stable process itself. A corresponding peaks-over-threshold approach will incorporate all single events that are extreme in some sense and will therefore rely on a substantially larger amount of data in comparison to estimation procedures based on block maxima. Conditioning a process \eta\ in the max-domain of attraction of \xi\ on being extremal, several convergence results for the increments of \eta\ are proved. In a similar way, the shape functions of mixed moving maxima (M3) processes can be extracted from suitably conditioned single events \eta. Connecting the two approaches, transformation formulae for processes that admit both an incremental and an M3 representation are identified.