On the consistency of inversion-free parameter estimation for Gaussian random fields

TL;DR

This paper investigates the asymptotic properties of a scalable, inversion-free algorithm for estimating covariance parameters of Gaussian random fields, demonstrating its consistency and optimality in high-dimensional settings.

Contribution

It provides the first theoretical analysis of the Anitescu, Chen, and Stein (ACS) algorithm, proving its consistency, minimax optimality, and asymptotic normality under mild conditions.

Findings

The ACS algorithm is consistent for regular and irregular grids.

It achieves minimax optimality in parameter estimation.

Numerical results confirm its efficiency for large datasets.

Abstract

Gaussian random fields are a powerful tool for modeling environmental processes. For high dimensional samples, classical approaches for estimating the covariance parameters require highly challenging and massive computations, such as the evaluation of the Cholesky factorization or solving linear systems. Recently, Anitescu, Chen and Stein \cite{M.Anitescu} proposed a fast and scalable algorithm which does not need such burdensome computations. The main focus of this article is to study the asymptotic behavior of the algorithm of Anitescu et al. (ACS) for regular and irregular grids in the increasing domain setting. Consistency, minimax optimality and asymptotic normality of this algorithm are proved under mild differentiability conditions on the covariance function. Despite the fact that ACS's method entails a non-concave maximization, our results hold for any stationary point of the objective function. A numerical study is presented to evaluate the efficiency of this algorithm for large data sets.