Abstract
We consider two Gaussian measures on a separable Hilbert space, with fractional-order covariance operators and , respectively, and derive necessary and sufficient conditions on and for I. equivalence of the measures μ and , and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure . These results hold, e.g., for Gaussian processes on compact metric spaces. As an important special case, we consider the class of generalized Whittle–Matérn Gaussian random fields, where A and are elliptic second-order differential operators, formulated on a bounded Euclidean domain and augmented with homogeneous Dirichlet boundary conditions. Our outcomes explain why the predictive performances of stationary and non-stationary models in spatial statistics often are comparable, and provide a crucial first step in deriving consistency results for parameter estimation of generalized Whittle–Matérn fields.
Acknowledgements
The authors thank the editor and the reviewers for their valuable comments which led to an improved, more accessible presentation of the results.
Citation
David Bolin. Kristin Kirchner. "Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators." Bernoulli 29 (2) 1476 - 1504, May 2023. https://doi.org/10.3150/22-BEJ1507