Open Access
June 2010 Adaptive estimation of stationary Gaussian fields
Nicolas Verzelen
Ann. Statist. 38(3): 1363-1402 (June 2010). DOI: 10.1214/09-AOS751


We study the nonparametric covariance estimation of a stationary Gaussian field X observed on a regular lattice. In the time series setting, some procedures like AIC are proved to achieve optimal model selection among autoregressive models. However, there exists no such equivalent results of adaptivity in a spatial setting. By considering collections of Gaussian Markov random fields (GMRF) as approximation sets for the distribution of X, we introduce a novel model selection procedure for spatial fields. For all neighborhoods m in a given collection $\mathcal{M}$, this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then it selects a neighborhood ̂m by applying a penalization strategy. The so-defined method satisfies a nonasymptotic oracle-type inequality. If X is a GMRF, the procedure is also minimax adaptive to the sparsity of its neighborhood. More generally, the procedure is adaptive to the rate of approximation of the true distribution by GMRFs with growing neighborhoods.


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Nicolas Verzelen. "Adaptive estimation of stationary Gaussian fields." Ann. Statist. 38 (3) 1363 - 1402, June 2010.


Published: June 2010
First available in Project Euclid: 8 March 2010

zbMATH: 1189.62157
MathSciNet: MR2662346
Digital Object Identifier: 10.1214/09-AOS751

Primary: 62H11
Secondary: 62M40

Keywords: Gaussian field , Gaussian Markov random field , minimax rate of estimation , Model selection , Oracle inequalities , pseudolikelihood

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 3 • June 2010
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