LARGE COVARIANCE ESTIMATION THROUGH ELLIPTICAL FACTOR MODELS

• 成果类型: Article
• 署名作者: Fan, Jianqing; Liu, Han; Wang, Weichen
• 署名单位: Princeton University; Fudan University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/17-AOS1588
• 发表日期: 2018
• 页码: 1383-1414
• 关键词: Principal component analysis; Precision Matrix Estimation; approximate factor models; Optimal Rates; sparse pca; statistical-analysis; quantile regression; Adaptive estimation; variable selection; Robust Estimation

摘要

We propose a general Principal Orthogonal complEment Thresholding (POET) framework for large-scale covariance matrix estimation based on the approximate factor model. A set of high-level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms is established to better understand how POET works. Such a framework allows us to recover existing results for sub-Gaussian data in a more transparent way that only depends on the concentration properties of the sample covariance matrix. As a new theoretical contribution, for the first time, such a framework allows us to exploit conditional sparsity covariance structure for the heavy-tailed data. In particular, for the elliptical distribution, we propose a robust estimator based on the marginal and spatial Kendall's tau to satisfy these conditions. In addition, we study conditional graphical model under the same framework. The technical tools developed in this paper are of general interest to high-dimensional principal component analysis. Thorough numerical results are also provided to back up the developed theory.