Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces

Abstract

Optimal linear prediction (aka. kriging) of a random field $\{Z(x){\}}_{x\in \mathcal{X}}$ indexed by a compact metric space $(\mathcal{X},d_{\mathcal{X}})$ can be obtained if the mean value function $m:\mathcal{X}\to \mathbb{R}$ and the covariance function $\varrho :\mathcal{X}\times \mathcal{X}\to \mathbb{R}$ of $Z$ are known. We consider the problem of predicting the value of $Z(x^{\ast })$ at some location $x^{\ast }\in \mathcal{X}$ based on observations at locations $\{x_j{\}}_{j=1}^n$ which accumulate at $x^{\ast }$ as $n\to \infty$ (or, more generally, predicting $\phi (Z)$ based on $\{{\phi }_j(Z){\}}_{j=1}^n$ for linear functionals $\phi ,{\phi }_1,\dots ,{\phi }_n$). Our main result characterizes the asymptotic performance of linear predictors (as $n$ increases) based on an incorrect second order structure $(\tilde{m},\tilde{\varrho })$, without any restrictive assumptions on $\varrho ,\tilde{\varrho }$ such as stationarity. We, for the first time, provide necessary and sufficient conditions on $(\tilde{m},\tilde{\varrho })$ for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to $\phi$. These general results are illustrated by weakly stationary random fields on $\mathcal{X}\subset {\mathbb{R}}^d$ with Matérn or periodic covariance functions, and on the sphere $\mathcal{X}={\mathbb{S}}^2$ for the case of two isotropic covariance functions.

36 pages (including the supplement). To appear in Annals of Statistics

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