Fixed-Domain Asymptotics Under Vecchia's Approximation of Spatial Process Likelihoods

TL;DR

This paper investigates the statistical properties of Vecchia's approximation for Gaussian spatial models in fixed-domain asymptotics, demonstrating its effectiveness for scalable inference on large spatial datasets.

Contribution

It establishes the inferential properties of covariance parameters estimated via Vecchia's approximation under fixed-domain asymptotics, both theoretically and empirically.

Findings

Vecchia's approximation retains key inferential properties in fixed-domain asymptotics.

Theoretical conditions for valid inference are identified and verified.

Empirical results support the approximation's effectiveness for large spatial data.

Abstract

Statistical modeling for massive spatial data sets has generated a substantial literature on scalable spatial processes based upon Vecchia's approximation. Vecchia's approximation for Gaussian process models enables fast evaluation of the likelihood by restricting dependencies at a location to its neighbors. We establish inferential properties of microergodic spatial covariance parameters within the paradigm of fixed-domain asymptotics when they are estimated using Vecchia's approximation. The conditions required to formally establish these properties are explored, theoretically and empirically, and the effectiveness of Vecchia's approximation is further corroborated from the standpoint of fixed-domain asymptotics.