Pseudo Best Estimator by a Separable Approximation of Spatial Covariance Structures

Abstract

We consider the linear regression model with a spatially correlated error term on a lattice process. When we estimate coefficients in the linear regression model, the generalized least squares estimator (GLSE) is used if the covariance structures are known. However, the GLSE for large spatial data sets is impractically time-consuming because it includes the inversion of the covariance matrix of error terms in different spatial points that is the size of the number of observations. To reduce the computational complexity, we propose the pseudo best estimator (PBE) using spatial covariance structures approximated by separable covariance functions. We derive the asymptotic covariance matrix of the PBE and compare it with those of the least squares estimator (LSE) and the GLSE through some simulations. They also imply that the effect of the misspecification of the covariance matrix for the GLSE is examined. Monte Carlo simulations demonstrate the improvement of the LSE, which does not contain the information of the spatial covariance structure, by the PBE using separable covariance functions even if the true process has an isotropic Matern covariance function. Additionally our proposed PBE is computationally efficient relative to the GLSE for large spatial data sets.