ADAPTIVE COVARIANCE MATRIX ESTIMATION THROUGH BLOCK THRESHOLDING

• 成果类型: Article
• 署名作者: Cai, Tony; Yuan, Ming
• 署名单位: University of Pennsylvania; University System of Georgia; Georgia Institute of Technology
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS999
• 发表日期: 2012
• 页码: 2014-2042
• 关键词: model selection; CONVERGENCE; likelihood; rates

摘要

Estimation of large covariance matrices has drawn considerable recent attention, and the theoretical focus so far has mainly been on developing a minimax theory over a fixed parameter space. In this paper, we consider adaptive covariance matrix estimation where the goal is to construct a single procedure which is minimax rate optimal simultaneously over each parameter space in a large collection. A fully data-driven block thresholding estimator is proposed. The estimator is constructed by carefully dividing the sample covariance matrix into blocks and then simultaneously estimating the entries in a block by thresholding. The estimator is shown to be optimally rate adaptive over a wide range of bandable covariance matrices. A simulation study is carried out and shows that the block thresholding estimator performs well numerically. Some of the technical tools developed in this paper can also be of independent interest.