Fixed and Increasing Domain Asymptotics for the Roughness and Scale of Isotropic Gaussian Random Fields

TL;DR

This paper develops a unified asymptotic theory for estimating roughness and scale parameters of isotropic Gaussian random fields, covering both fixed and increasing domain scenarios, with explicit results and robustness to sampling irregularities.

Contribution

It provides the first unified distributional framework for joint estimation in Gaussian fields under both asymptotic regimes, including explicit formulas and robustness extensions.

Findings

Bivariate CLTs for estimators under both regimes

Explicit expressions for roughness and scale estimates

Robustness to irregular sampling and model extensions

Abstract

We establish a rigorous asymptotic theory for the joint estimation of roughness and scale parameters in two-dimensional Gaussian random fields with power-law generalized covariances \cite{Matheron1973, Stein1999, Yaglom1987}. Our main results are bivariate central limit theorems for a class of method-of-moments estimators under increasing-domain and fixed-domain asymptotics. The fixed-domain result follows immediately from the increasing-domain result from the self-similarity of Gaussian random fields with power-law generalized covariances \cite{IstasLang1997, Coeurjolly2001, ZhuStein2002}. These results provide a unified distributional framework across these two classical regimes \cite{AvramLeonenkoSakhno2010-ESAIM, BiermeBonamiLeon2011-EJP} that makes the unusual behavior of the estimates under fixed-domain asymptotics intuitively obvious. Our increasing-domain asymptotic results use spatial averages of quadratic forms of (iterated) bilinear product difference filters that yield explicit expressions for the estimates of roughness and scale to which existing theorems on such averages \cite{BreuerMajor1983,Hannan1970} can be readily applied. We further show that the asymptotics remain valid under modestly irregular sampling due to jitter or missing observations. For the fixed-domain setting, the results extend to models that behave sufficiently like the power-law model at high frequencies such as the often used Mat\'ern model \cite{ZhuStein2006, WangLoh2011EJS, KaufmanShaby2017EJS}.