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

Nicolas Chenavier, Christian Y. Robert

TL;DR

This work develops fixed-domain (infill) asymptotics for maximum composite likelihood estimators (MCLE) of the scale and Hurst parameters in Brown-Resnick max-stable fields tied to isotropic fractional Brownian fields. By embedding a randomized Poisson sampling within a fixed domain and using the Delaunay triangulation to select informative pairs and triples, the authors derive asymptotic properties of pairwise and triplewise CL objectives; the limiting behavior is driven by local times of the underlying process rather than classical Gaussian limits. They show consistency of the MCLEs and establish nonstandard convergence rates tied to the local-time structure, with explicit score-function expansions for both pairwise and triplewise cases. These results provide theoretical guidance on inference for spatial extreme value models in a single realization setting and highlight the role of tessellations and local times in non-Gaussian fixed-domain asymptotics. The practical impact lies in enabling principled parameter estimation for max-stable spatial models when data are precious or observations are limited to a fixed region.

Abstract

Likelihood inference for max-stable random fields is in general impossible because their finite-dimensional 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.

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

TL;DR

This work develops fixed-domain (infill) asymptotics for maximum composite likelihood estimators (MCLE) of the scale and Hurst parameters in Brown-Resnick max-stable fields tied to isotropic fractional Brownian fields. By embedding a randomized Poisson sampling within a fixed domain and using the Delaunay triangulation to select informative pairs and triples, the authors derive asymptotic properties of pairwise and triplewise CL objectives; the limiting behavior is driven by local times of the underlying process rather than classical Gaussian limits. They show consistency of the MCLEs and establish nonstandard convergence rates tied to the local-time structure, with explicit score-function expansions for both pairwise and triplewise cases. These results provide theoretical guidance on inference for spatial extreme value models in a single realization setting and highlight the role of tessellations and local times in non-Gaussian fixed-domain asymptotics. The practical impact lies in enabling principled parameter estimation for max-stable spatial models when data are precious or observations are limited to a fixed region.

Abstract

Likelihood inference for max-stable random fields is in general impossible because their finite-dimensional 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 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.

Paper Structure

This paper contains 35 sections, 8 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Maximum likelihood estimators for Gaussian random fields under fixed-domain asymptotics
  3. Maximum composite likelihood estimators for Gaussian random fields under fixed-domain asymptotics
  4. Fixed-domain asymptotics for non-Gaussian random fields
  5. Max-stable random fields
  6. Contributions of the paper
  7. Preliminaries
  8. Max-stable Brown-Resnick random fields
  9. Delaunay triangulation
  10. Local time
  11. Increments of a Brown-Resnick random field
  12. Asymptotic distributions of normalized increments
  13. Single increments
  14. Pairs of increments
  15. Asymptotic distributions of squared increment sums for the max-stable Brown-Resnick random field
  16. ...and 20 more sections

Key Result

Proposition 1

Let $x_1$, $x_2$ be two (distinct) points in $\mathbf{R}^2$ and $d_{1,2}=||x_2-x_1||$.

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Remark 1
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Remark 2
  • Lemma 7
  • Proposition 8