Fixed-domain asymptotic properties of maximum composite likelihood estimators for max-stable Brown-Resnick random fields

TL;DR

This paper investigates the asymptotic behavior of maximum composite likelihood estimators for max-stable Brown-Resnick fields, focusing on the effect of different weighting strategies as the observation intensity grows.

Contribution

It provides new theoretical insights into the fixed-domain asymptotics of composite likelihood estimators for Brown-Resnick fields with various weighting schemes.

Findings

Asymptotic properties depend on the weighting strategy used.

Excluding non-edge pairs or non-triangle triples affects estimator consistency.

Results inform optimal weighting choices for spatial max-stable field inference.

Abstract

Likelihood inference for max-stable random fields is in general impossible because their finite-dimen\-sional probability density functions are unknown or cannot be computed efficiently. The weighted composite likelihood approach that utilizes lower dimensional marginal likelihoods (typically pairs or triples of sites that are not too distant) is rather favored. In this paper, we consider the family of spatial max-stable Brown-Resnick random fields associated with isotropic fractional Brownian fields. We assume that the sites are given by only one realization of a homogeneous Poisson point process restricted to $\mathbf{C}=(-1/2,1/2]^{2}$ and that the random field is observed at these sites. As the intensity increases, we study the asymptotic properties of the composite likelihood estimators of the scale and Hurst parameters of the fractional Brownian fields using different weighting strategies: we exclude either pairs that are not edges of the Delaunay triangulation or triples that are not vertices of triangles.