Estimation of spatial max-stable models using threshold exceedances

TL;DR

This paper proposes and compares two likelihood-based methods for estimating spatial max-stable models using threshold exceedances, offering a potentially more efficient alternative to block maxima approaches, especially with high-frequency data.

Contribution

It introduces two novel likelihood-based estimation approaches for spatial max-stable models using exceedances, improving inference with high-frequency data over traditional block maxima methods.

Findings

Both methods perform well depending on spatial dependency and threshold choice.

The threshold and dependency strength significantly influence estimation accuracy.

Simulation results guide the choice of method based on data characteristics.

Abstract

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.