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.

翻译：椭圆因子模型在现代高维数据分析中占据核心地位，尤其因其能够捕捉重尾和异质性依赖结构而备受关注。在此框架下，Tyler的M估计量（Tyler, 1987a）具有若干最优性质与稳健性优势。本文基于Tyler的M估计，发展了高维散度矩阵、协方差矩阵和精度矩阵的估计方法。我们首先通过引入空间符号协方差矩阵作为有效的初始估计量，对主正交补阈值法（POET）框架（Fan et al., 2013）进行适应性改进。基于这一思路，我们进一步提出了专为Tyler的M估计量身定制的POET直接扩展方法，称为POET-TME方法。我们在椭圆因子模型下建立了所得估计量的一致性收敛速率。全面的模拟研究和实际数据应用表明，POET-TME方法具有优越性能，尤其在存在重尾分布的情况下，这证明了我们方法学贡献的实用价值。