An h-likelihood method for spatial mixed linear models based on intrinsic auto-regressions

We consider sparse spatial mixed linear models, particularly those described by Besag and Higdon, and develop an h-likelihood method for their statistical inference. The method proposed allows for singular precision matrices, as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model. Furthermore, we generalize the h-likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between Gaussian intrinsic Markov random fields on regular arrays and the de Wijs process. Keeping various applications of spatial mixed linear models in mind, we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix-free statistical computations. We provide two applications. The first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation. The second application concerns an analysis of a large cotton field which gives a focus to matrix-free computations. The paper closes with some further considerations, such as applications to irregularly spaced data, use of the parametric bootstrap and some generalizations to the Gaussian Matern mixed effect models.