On the asymptotics of maximum likelihood estimation for spatial linear models on a lattice

Spatial linear models and the corresponding likelihood-based statistical inference are important tools for the analysis of spatial lattice data and have been applied in a wide range of disciplines. However, understanding of the asymptotic properties of maximum likelihood estimates is limited. Here we consider a unified asymptotic framework that encompasses increasing domain, infill, and a combination of increasing domain and infill asymptotics. Under each type of asymptotics, we derive the asymptotic properties of maximum likelihood estimates. Our results show that the rates of convergence vary for different asymptotic types and under infill asymptotics, some of the model parameters estimates are inconsistent. A simulation study is conducted to examine the finite-sample properties of the maximum likelihood estimates.