Asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields over a fixed-domain

Abstract

Likelihood-based inference for max-stable random fields is challenging, since finite-dimensional densities are either unavailable in closed form or computationally intractable in moderate to high dimension. Composite likelihood methods, based on low-dimensional marginal densities, therefore provide a natural alternative. In this paper, we study maximum composite likelihood estimation for spatial Brown--Resnick random fields generated by isotropic fractional Brownian fields. We work under fixed-domain asymptotics: a single realization of the max-stable field is observed on an increasingly dense random set of sites, given by a homogeneous Poisson point process. Pairwise and triplewise composite likelihoods are constructed by retaining, respectively, the edges and the triangles of the associated Poisson--Delaunay triangulation. Our main results establish the consistency of the resulting maximum composite likelihood estimators of the scale and smoothness parameters, when the other parameter is known. Their asymptotic behaviour is non-standard: the estimators converge at rates depending on the smoothness parameter and their centered limits are non-Gaussian. More precisely, the limiting fluctuations are driven by aggregated local times associated with the canonical tessellation of the Brown--Resnick field. These results reveal a fundamental departure from the classical composite likelihood theory based on increasing domains or independent replications, and show that Gaussian uncertainty quantification may be misleading in fixed-domain inference for max-stable spatial extremes.