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Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics


Bevilacqua, M; Faouzi, T; Furrer, Reinhard; Porcu, E (2019). Estimation and prediction using generalized Wendland covariance functions under fixed domain asymptotics. The Annals of Statistics, 47(2):825-856.

Abstract

We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As for the Matérn case, this class allows for a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into three parts: first, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. The third part elucidates the consequences of our results in terms of (misspecified) best linear unbiased predictor, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the former compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. The latter compares the finite-sample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance models, using covariance tapering as benchmark.

Abstract

We study estimation and prediction of Gaussian random fields with covariance models belonging to the generalized Wendland (GW) class, under fixed domain asymptotics. As for the Matérn case, this class allows for a continuous parameterization of smoothness of the underlying Gaussian random field, being additionally compactly supported. The paper is divided into three parts: first, we characterize the equivalence of two Gaussian measures with GW covariance function, and we provide sufficient conditions for the equivalence of two Gaussian measures with Matérn and GW covariance functions. In the second part, we establish strong consistency and asymptotic distribution of the maximum likelihood estimator of the microergodic parameter associated to GW covariance model, under fixed domain asymptotics. The third part elucidates the consequences of our results in terms of (misspecified) best linear unbiased predictor, under fixed domain asymptotics. Our findings are illustrated through a simulation study: the former compares the finite sample behavior of the maximum likelihood estimation of the microergodic parameter with the given asymptotic distribution. The latter compares the finite-sample behavior of the prediction and its associated mean square error when using two equivalent Gaussian measures with Matérn and GW covariance models, using covariance tapering as benchmark.

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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 for Computational Science
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2019
Deposited On:28 Jun 2019 12:52
Last Modified:14 Feb 2020 15:38
Publisher:Institute of Mathematical Statistics
ISSN:0090-5364
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/17-AOS1652
Official URL:https://projecteuclid.org/euclid.aos/1547197240

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