Penultimate modeling of spatial extremes: statistical inference for max-infinitely divisible processes

Abstract

Extreme-value theory for stochastic processes has motivated the statistical use of max-stable models for spatial extremes. However, fitting such asymptotic models to maxima observed over finite blocks is problematic when the asymptotic stability of the dependence does not prevail in finite samples. This issue is particularly serious when data are asymptotically independent, such that the dependence strength weakens and eventually vanishes as events become more extreme. We here aim to provide flexible sub-asymptotic models for spatially indexed block maxima, which more realistically account for discrepancies between data and asymptotic theory. We develop models pertaining to the wider class of max-infinitely divisible processes, extending the class of max-stable processes while retaining dependence properties that are natural for maxima: max-id models are positively associated, and they yield a self-consistent family of models for block maxima defined over any time unit. We propose two parametric construction principles for max-id models, emphasizing a point process-based generalized spectral representation, that allows for asymptotic independence while keeping the max-stable extremal-$t$ model as a special case. Parameter estimation is efficiently performed by pairwise likelihood, and we illustrate our new modeling framework with an application to Dutch wind gust maxima calculated over different time units.