High dimensional matrix estimation through elliptical factor models

TL;DR

This paper introduces a new high-dimensional covariance estimation method based on Tyler's M-estimator within elliptical factor models, demonstrating improved robustness and accuracy in heavy-tailed data scenarios.

Contribution

It extends the POET framework by integrating Tyler's M-estimator, resulting in a novel estimator called POET-TME with proven consistency and superior performance.

Findings

POET-TME outperforms existing methods in simulations with heavy-tailed data.

The proposed estimators are consistent under elliptical factor models.

Real data analysis confirms practical advantages of POET-TME.

Abstract

Elliptical factor models play a central role in modern high-dimensional data analysis, particularly due to their ability to capture heavy-tailed and heterogeneous dependence structures. Within this framework, Tyler's M-estimator (Tyler, 1987a) enjoys several optimality properties and robustness advantages. In this paper, we develop high-dimensional scatter matrix, covariance matrix and precision matrix estimators grounded in Tyler's M-estimation. We first adapt the Principal Orthogonal complEment Thresholding (POET) framework (Fan et al., 2013) by incorporating the spatial-sign covariance matrix as an effective initial estimator. Building on this idea, we further propose a direct extension of POET tailored for Tyler's M-estimation, referred to as the POET-TME method. We establish the consistency rates for the resulting estimators under elliptical factor models. Comprehensive simulation studies and a real data application illustrate the superior performance of POET-TME, especially in the presence of heavy-tailed distributions, demonstrating the practical value of our methodological contributions.