Asymptotically Equivalent Prediction in Multivariate Geostatistics

TL;DR

This paper investigates the conditions under which multivariate cokriging predictions are asymptotically equivalent, focusing on Gaussian measures and covariance functions, to improve understanding of spatial interpolation in multivariate geostatistics.

Contribution

It provides the first comprehensive conditions for measure equivalence in multivariate Gaussian fields, especially for Matérn and Wendland classes, addressing a long-standing gap in spatial statistics.

Findings

Conditions for measure equivalence in multivariate Gaussian fields.

Identification of key covariance parameters affecting asymptotic prediction.

Simulation studies validating theoretical results.

Abstract

Cokriging is the common method of spatial interpolation (best linear unbiased prediction) in multivariate geostatistics. While best linear prediction has been well understood in univariate spatial statistics, the literature for the multivariate case has been elusive so far. The new challenges provided by modern spatial datasets, being typically multivariate, call for a deeper study of cokriging. In particular, we deal with the problem of misspecified cokriging prediction within the framework of fixed domain asymptotics. Specifically, we provide conditions for equivalence of measures associated with multivariate Gaussian random fields, with index set in a compact set of a d-dimensional Euclidean space. Such conditions have been elusive for over about 50 years of spatial statistics. We then focus on the multivariate Mat\'ern and Generalized Wendland classes of matrix valued covariance functions, that have been very popular for having parameters that are crucial to spatial interpolation, and that control the mean square differentiability of the associated Gaussian process. We provide sufficient conditions, for equivalence of Gaussian measures, relying on the covariance parameters of these two classes. This enables to identify the parameters that are crucial to asymptotically equivalent interpolation in multivariate geostatistics. Our findings are then illustrated through simulation studies.