High dimensional matrix estimation through elliptical factor models

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.