Open Access
August 2018 Large covariance estimation through elliptical factor models
Jianqing Fan, Han Liu, Weichen Wang
Ann. Statist. 46(4): 1383-1414 (August 2018). DOI: 10.1214/17-AOS1588


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.


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Jianqing Fan. Han Liu. Weichen Wang. "Large covariance estimation through elliptical factor models." Ann. Statist. 46 (4) 1383 - 1414, August 2018.


Received: 1 July 2015; Revised: 1 January 2017; Published: August 2018
First available in Project Euclid: 27 June 2018

zbMATH: 06936465
MathSciNet: MR3819104
Digital Object Identifier: 10.1214/17-AOS1588

Primary: 62H25
Secondary: 62H12

Keywords: approximate factor model , conditional graphical model , elliptical distribution , marginal and spatial Kendall’s tau , Principal Component Analysis , sub-Gaussian family

Rights: Copyright © 2018 Institute of Mathematical Statistics

Vol.46 • No. 4 • August 2018
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