Large covariance estimation through elliptical factor models

Jianqing Fan, Han Liu, Weichen Wang

Research output: Contribution to journalArticlepeer-review

44 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1383-1414
Number of pages32
JournalAnnals of Statistics
Issue number4
StatePublished - Aug 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Approximate factor model
  • Conditional graphical model
  • Elliptical distribution
  • Marginal
  • Principal component analysis
  • Spatial Kendall’s tau
  • Sub-Gaussian family


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