Open Access
April 2022 Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces
Kristin Kirchner, David Bolin
Author Affiliations +
Ann. Statist. 50(2): 1038-1065 (April 2022). DOI: 10.1214/21-AOS2138


Optimal linear prediction (aka. kriging) of a random field {Z(x)}xX indexed by a compact metric space (X,dX) can be obtained if the mean value function m:XR and the covariance function ϱ:X×XR of Z are known. We consider the problem of predicting the value of Z(x) at some location xX based on observations at locations {xj}j=1n, which accumulate at x as n (or, more generally, predicting φ(Z) based on {φj(Z)}j=1n for linear functionals φ,φ1,,φn). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure (m˜,ϱ˜), without any restrictive assumptions on ϱ, ϱ˜ such as stationarity. We, for the first time, provide necessary and sufficient conditions on (m˜,ϱ˜) for asymptotic optimality of the corresponding linear predictor holding uniformly with respect to φ. These general results are illustrated by weakly stationary random fields on XRd with Matérn or periodic covariance functions, and on the sphere X=S2 for the case of two isotropic covariance functions.


The authors thank S.G. Cox and J.M.A.M. van Neerven for fruitful discussions on spectral theory, which considerably contributed to the proof of Lemma B.2; see Appendix B in the Supplementary Material [9]. In addition, we thank the Editor and an anonymous reviewer for their valuable comments.


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Kristin Kirchner. David Bolin. "Necessary and sufficient conditions for asymptotically optimal linear prediction of random fields on compact metric spaces." Ann. Statist. 50 (2) 1038 - 1065, April 2022.


Received: 1 May 2020; Revised: 1 September 2021; Published: April 2022
First available in Project Euclid: 7 April 2022

MathSciNet: MR4404928
zbMATH: 1486.62325
Digital Object Identifier: 10.1214/21-AOS2138

Primary: 62M20
Secondary: 60G12 , 60G25 , 60G60

Keywords: Approximation in Hilbert spaces , kriging , spatial statistics

Rights: Copyright © 2022 Institute of Mathematical Statistics

Vol.50 • No. 2 • April 2022
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