Abstract
Optimal linear prediction (aka. kriging) of a random field indexed by a compact metric space can be obtained if the mean value function and the covariance function of Z are known. We consider the problem of predicting the value of at some location based on observations at locations , which accumulate at as (or, more generally, predicting based on for linear functionals ). Our main result characterizes the asymptotic performance of linear predictors (as n increases) based on an incorrect second-order structure , without any restrictive assumptions on ϱ, such as stationarity. We, for the first time, provide necessary and sufficient conditions on 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 with Matérn or periodic covariance functions, and on the sphere for the case of two isotropic covariance functions.
Acknowledgments
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.
Citation
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. https://doi.org/10.1214/21-AOS2138
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