On the consistency of inversion-free parameter estimation for Gaussian random fields
This work addresses the computational bottleneck in environmental modeling for researchers and practitioners, but it is incremental as it builds on an existing algorithm.
The paper tackles the problem of computationally expensive parameter estimation for Gaussian random fields by analyzing the asymptotic behavior of an inversion-free algorithm, proving consistency, minimax optimality, and asymptotic normality under mild conditions.
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.