Large Covariance Estimation through Elliptical Factor Models

TL;DR

This paper introduces a flexible framework for large covariance matrix estimation using elliptical factor models, incorporating robust methods for heavy-tailed data and providing theoretical and numerical validation.

Contribution

It develops a general POET framework that achieves optimal convergence rates and extends to elliptical data with robust estimators, advancing high-dimensional covariance estimation.

Findings

Achieves optimal convergence rates under various norms.

Introduces a robust estimator for elliptical data using Kendall's tau.

Provides numerical results validating the theoretical developments.

Abstract

We proposed a general Principal Orthogonal complEment Thresholding (POET) framework for large-scale covariance matrix estimation based on an approximate factor model. A set of high level sufficient conditions for the procedure to achieve optimal rates of convergence under different matrix norms were brought up to better understand how POET works. Such a framework allows us to recover the results for sub-Gaussian 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 data, we proposed a robust estimator based on marginal and multivariate Kendall's tau to satisfy these conditions. In addition, conditional graphical model was also studied under the same framework. The technical tools developed in this paper are of general interest to high dimensional principal component analysis. Thorough numerical results were also provided to back up the developed theory.