Statistical modeling 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 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. These explorations suggest some practical diagnostics for evaluating the quality of the approximation.

翻译：大量空间数据集的统计模型产生了大量文献,说明根据Vecchia1988年提出的可能性近似值可扩展的空间过程。Vecchia对Gaussian过程模型的近似值能够通过将一个地点的依赖性限制在邻近地区来快速评估可能性。我们在使用Vecchia的近近似值估算固定地貌时,在固定地表空间多变参数的范式范围内,确定微遗传空间常变参数的推论特性。从固定地貌的零现性的角度探讨正式确定这些特性所需的条件,并从理论上和经验上进一步证实Vecchia的近似值的有效性。这些探索表明,在评估近似值质量时,可以进行一些实际的诊断。