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Max-stable processes are the most popular models for high-impact spatialextreme events, as they arise as the only possible limits of spatially-indexedblock maxima. However, likelihood inference for such models suffers severelyfrom the curse of dimensionality, since the likelihood function involves acombinatorially exploding number of terms. In this paper, we propose using theVecchia approximation, which conveniently decomposes the full joint densityinto a linear number of low-dimensional conditional density terms based onwell-chosen conditioning sets designed to improve and accelerate inference inhigh dimensions. Theoretical asymptotic relative efficiencies in the Gaussiansetting and simulation experiments in the max-stable setting show significantefficiency gains and computational savings using the Vecchia likelihoodapproximation method compared to traditional composite likelihoods. Ourapplication to extreme sea surface temperature data at more than a thousandsites across the entire Red Sea further demonstrates the superiority of theVecchia likelihood approximation for fitting complex models with intractablelikelihoods, delivering significantly better results than traditional compositelikelihoods, and accurately capturing the extremal dependence structure atlower computational cost.

Vecchia Likelihood Approximation for Accurate and Fast Inference in Intractable Spatial Extremes Models