Estimation of Functionals of High Dimensional Covariance Matrices

在涉及协方差的广泛应用中，人们对协方差结构的某些方面，即泛函感兴趣。尽管近年来在协方差矩阵估计的方法学研究上取得了进展，但关于高维协方差矩阵泛函的最优估计的基础理论和方法学研究却非常少。本提案的目标是发展一个连贯的理论，以揭示协方差矩阵函数可以估计的精度，开发协方差矩阵函数的最佳估计的一般方法，建立协方差矩阵估计与高斯序列模型之间的渐近等价，并解决在金融，生物信息学，基因组学和气象学等领域出现的应用。本文的研究将极大地促进对大规模协方差矩阵泛函估计的理论认识。特别是，即将发展的渐近等价理论将有助于在矩阵估计和经典高斯序列模型之间建立密切和鼓舞人心的联系，这在过去三十年中已经得到了很好的研究，并有助于将高斯序列模型中开发的结果和方法转移到矩阵估计中。对于那些重要的统计方法，如主成分分析、图形模型、线性和二次判别分析，提出了最优估计程序，将在广泛的应用中提供更准确的估计和预测规则。随着现代技术带来的高维数据的出现，大规模协方差矩阵及其函数的估计已成为气候研究、基因组学和蛋白质组学、功能磁共振成像、投资组合配置和风险管理等诸多领域的关键问题。传统的样本协方差矩阵估计在实际分析高维数据时经常被使用，可能导致性能差和结论无效。为了克服高维的困难，近年来发展了正则化方法。协方差矩阵在统计分析中的核心作用及其广泛的重要统计应用确保了研究者和他的同事在他们提出的目标方面取得的进展将在广泛的科学界产生重大影响，包括天文学、生物信息学、金融、基因组学、气象学和临床研究。这项建议的研究成果将通过研究文章、讲习班和研讨会系列传播给其他学科的研究人员。该项目将通过教授专题课程和组织讲习班和研讨会来整合研究和教育，以帮助研究生和博士后，特别是少数民族、妇女、国内学生和年轻研究人员研究这一主题。

In a wide range of applications involving covariance, people are interested in certain aspects, i.e., functionals, of the covariance structure. Despite recent progress on methodological work on covariance matrices estimation, there has been remarkably little fundamental theoretical and methodological studies on optimal estimation of functionals of high dimensional covariance matrices. The goal of this proposal is to develop a coherent theory to unveil the precision to which covariance matrix functionals can be estimated, to develop general methodologies for optimal estimation of functionals of covariance matrices, to establish the asymptotic equivalence between covariance matrices estimation and Gaussian sequence models, and to address applications that arise in finance, bioinformatics, genomics, and meteorology etc. The research presented in this proposal will significantly advance the theoretical understanding of estimating functionals of large scale covariance matrices. In particular, the asymptotic equivalence theory to be developed will help build a close and inspiring connection between matrices estimation and classical Gaussian sequence models which had been well studied in the past thirty years, and help carry over results and methodologies developed in Gaussian sequence models to matrices estimation. The proposed optimal estimation procedures for those fundamentally important statistics methodologies, for example, principal components analysis, graphical model, and linear and quadratic discriminant analysis, will provide more accurate estimation and prediction rules in a wide range of applications.With the emergence of high dimensional data from modern technologies, estimating large scale covariance matrices and their functionals is becoming a crucial problem in many fields including climate studies, genomics and proteomics, functional magnetic resonance imaging, portfolio allocation and risk management. The traditional sample covariance matrix estimator has been used frequently in practice when analyzing high dimensional data, which may result in poor performance and invalid conclusions. To overcome the difficulty associated with the high dimensionality, regularized methods have been developed in recent years. A central role of covariance matrix in statistical analysis and its wide range of important statistical applications ensure that the progress the investigator and his colleagues make towards their proposed objectives will have a great impact in the broad scientific community which includes astronomy, bioinformatics, finance, genomics, meteorology and clinical research. Research results from this proposal will be disseminated through research articles, workshops and seminar series to researchers in other disciplines. The project will integrate research and education by teaching monograph courses and organizing workshops and seminars to help graduate students and postdocs, particularly minority, women, and domestic students and young researchers, work on this topic.