ADAPTIVE ESTIMATION OF STATIONARY GAUSSIAN FIELDS

• 成果类型: Article
• 署名作者: Verzelen, Nicolas
• 署名单位: INRAE; Institut Agro; Montpellier SupAgro; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS751
• 发表日期: 2010
• 页码: 1363-1402
• 关键词: markov random-fields; selection; models; inequalities; ORDER

摘要

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 M, this procedure first amounts to computing a covariance estimator of X within the GMRFs of neighborhood m. Then it selects a neighborhood (m) over cap 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.