Unifying compactly supported and Matérn covariance functions in spatial statistics

The Matérn family of covariance functions has played a central role in spatial statistics for decades, being a flexible parametric class with one parameter determining the smoothness of the paths of the underlying spatial field. This paper proposes a family of spatial covariance functions, which stems from a reparameterization of the generalized Wendland family. As for the Matérn case, the proposed family allows for a continuous parameterization of the smoothness of the underlying Gaussian random field, being additionally compactly supported. More importantly, we show that the proposed covariance family generalizes the Matérn model which is attained as a special limit case. This implies that the (reparametrized) Generalized Wendland model is more flexible than the Matérn model with an extra-parameter that allows for switching from compactly to globally supported covariance functions. Our numerical experiments elucidate the speed of convergence of the proposed model to the Matérn model. We also inspect the asymptotic distribution of the maximum likelihood method when estimating the parameters of the proposed covariance models under both increasing and fixed domain asymptotics. The effectiveness of our proposal is illustrated by analyzing a georeferenced dataset of mean temperatures over a region of French, and performing a re-analysis of a large spatial point referenced dataset of yearly total precipitation anomalies.