Estimation of spatial max-stable models using threshold exceedances

Parametric inference for spatial max-stable processes is difficult since the related likelihoods are unavailable. A composite likelihood approach based on the bivariate distribution of block maxima has been recently proposed in the literature. However modeling block maxima is a wasteful approach provided that other information is available. Moreover an approach based on block, typically annual, maxima is unable to take into account the fact that maxima occur or not simultaneously. If time series of, say, daily data are available, then estimation procedures based on exceedances of a high threshold could mitigate such problems. In this paper we focus on two approaches for composing likelihoods based on pairs of exceedances. The first one comes from the tail approximation for bivariate distribution proposed by Ledford and Tawn (1996) when both pairs of observations exceed the fixed threshold. The second one uses the bivariate extension (Rootzen and Tajvidi, 2006) of the generalized Pareto distribution which allows to model exceedances when at least one of the components is over the threshold. The two approaches are compared through a simulation study according to different degrees of spatial dependency. Results show that both the strength of the spatial dependencies and the threshold choice play a fundamental role in determining which is the best estimating procedure.

💡 Research Summary

The paper addresses a fundamental difficulty in the statistical inference of spatial max‑stable processes: the full joint likelihood is unavailable because the multivariate density of max‑stable distributions does not have a closed form. Traditionally, researchers have resorted to composite likelihood methods that use the bivariate distribution of block maxima (usually annual maxima). While mathematically convenient, this approach discards a large amount of information contained in the daily (or higher‑frequency) records and cannot capture the fact that extreme events may or may not occur simultaneously at different sites.

To overcome these limitations, the authors propose two composite‑likelihood constructions that are based on exceedances over a high threshold rather than on block maxima. Both methods operate on pairs of sites, preserving the computational tractability of pairwise likelihoods while exploiting the richer information in threshold exceedances.

1. Tail‑approximation approach (Ledford & Tawn, 1996).
When both observations in a pair exceed the fixed threshold (u), the joint tail can be approximated by
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