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Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations


Furrer, Reinhard; Hediger, Michael (2023). Asymptotic analysis of ML-covariance parameter estimators based on covariance approximations. Electronic Journal of Statistics, 17(2):3050-3102.

Abstract

Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncatedlikelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions.
For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.

Abstract

Given a zero-mean Gaussian random field with a covariance function that belongs to a parametric family of covariance functions, we introduce a new notion of likelihood approximations, termed truncatedlikelihood functions. Truncated-likelihood functions are based on direct functional approximations of the presumed family of covariance functions.
For compactly supported covariance functions, within an increasing-domain asymptotic framework, we provide sufficient conditions under which consistency and asymptotic normality of estimators based on truncated-likelihood functions are preserved. We apply our result to the family of generalized Wendland covariance functions and discuss several examples of Wendland approximations. For families of covariance functions that are not compactly supported, we combine our results with the covariance tapering approach and show that ML estimators, based on truncated-tapered likelihood functions, asymptotically minimize the Kullback-Leibler divergence, when the taper range is fixed.

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Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
07 Faculty of Science > Institute of Evolutionary Biology and Environmental Studies
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics and Probability, Statistics, Probability and Uncertainty 60G15 - Gaussian processes 41 - Approximations and expansions 62F12 - Asymptotic properties of parametric estimators 62M40 - Random fields; image analysis Primary 60G15, 62F12; secondary 41A99. Keywords and phrases: Gaussian random fields, compactly supported covariance functions, likelihood approximations, consistency, asymptotic normality, covariance tapering.
Language:English
Date:1 January 2023
Deposited On:20 Dec 2023 10:01
Last Modified:27 Jun 2024 03:43
Publisher:Institute of Mathematical Statistics
ISSN:1935-7524
Additional Information:Acknowledgments: The authors thank Roman Flury for all the stimulating discussions that were held during the development of this work. Funding: This work was supported by the Swiss National Science Foundation SNSF-175529.
OA Status:Gold
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/23-ejs2170
Related URLs:https://www.zora.uzh.ch/id/eprint/230502/
https://doi.org/10.48550/arXiv.2112.12317
Other Identification Number:MR4667731
Project Information:
  • : FunderSwiss National Science Foundation
  • : Grant IDSNSF-175529
  • : Project TitleSNSF-175529
  • Content: Published Version
  • Language: English
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)