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

Statistical modelling for massive spatial data sets has generated a substantial literature on scalable spatial processes based upon a likelihood approximation proposed by Vecchia in 1988. Vecchia's approximation for Gaussian process models enables fast evaluation of the likelihood by restricting dependencies at a location to its neighbours. 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. These explorations suggest some practical diagnostics for evaluating the quality of the approximation.