Large covariance estimation through elliptical factor models

2018/06/27 by Jianqing Fan, Han Liu, Weichen Wang · 103 citations
Mathematics · #Advanced Statistical Methods and Models #Algorithm #Applied mathematics #Complement (music) #Convergence (economics) #Covariance #Covariance function #Covariance matrix #Elliptical distribution #Estimation of covariance matrices #Estimator #Factor analysis #Gaussian #Mathematical optimization #Mathematics #Matrix (chemical analysis) #Multivariate normal distribution #Multivariate statistics #Principal component analysis #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics

paper · pdf · doi:10.1214/17-aos1588

published in The Annals of Statistics 46(4), 1383-1414 (Institute of Mathematical Statistics)

openalex publication_date 2018/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

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.

Citations

• The Rotation of Eigenvectors by a Perturbation. III
• Operator norm consistent estimation of large-dimensional sparse covariance matrices
• Robust Estimation of a Location Parameter
• Optimal rates of convergence for covariance matrix estimation
• Statistical analysis of factor models of high dimension

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• Large Covariance Estimation through Elliptical Factor Models
• Large Covariance Estimation by Thresholding Principal Orthogonal Complements
• High dimensional matrix estimation through elliptical factor models
• Sub-Gaussian High-Dimensional Covariance Matrix Estimation under Elliptical Factor Model with 2 + εth Moment
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