Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

Abstract

We consider Gaussian measures $\mu ,\tilde{\mu }$ on a separable Hilbert space, with fractional-order covariance operators $A^{-2\beta }$ resp. ${\tilde{A}}^{-2\tilde{\beta }}$, and derive necessary and sufficient conditions on $A,\tilde{A}$ and $\beta ,\tilde{\beta }>0$ for I. equivalence of the measures $\mu$ and $\tilde{\mu }$, and II. uniform asymptotic optimality of linear predictions for $\mu$ based on the misspecified measure $\tilde{\mu }$. 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 $\tilde{A}$ are elliptic second-order differential operators, formulated on a bounded Euclidean domain $\mathcal{D}\subset {\mathbb{R}}^d$ 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.

47 pages, 3 figures

References in corpus (4)

• Spatial models generated by nested stochastic partial differential equations, with an application to global ozone mapping
• Matérn Gaussian processes on Riemannian manifolds
• Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
• Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators

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• Smoothness Estimation for Whittle-Matérn Processes on Closed Riemannian Manifolds