Equivalence of measures and asymptotically optimal linear prediction for Gaussian random fields with fractional-order covariance operators
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We consider Gaussian measures μ, μ̃ on a separable Hilbert space, with fractional-order covariance operators A^-2β resp. Ã^-2β̃, and derive necessary and sufficient conditions on A, à and β, β̃ > 0 for I. equivalence of the measures μ and μ̃, and II. uniform asymptotic optimality of linear predictions for μ based on the misspecified measure μ̃. 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 à are elliptic second-order differential operators, formulated on a bounded Euclidean domain 𝒟⊂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.
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