Sub-Gaussian High-Dimensional Covariance Matrix Estimation under Elliptical Factor Model with 2 + {\epsilon}th Moment

TL;DR

This paper introduces a novel robust covariance matrix estimator for high-dimensional elliptical factor models with heavy-tailed data, achieving sub-Gaussian convergence rates where previous methods failed.

Contribution

It develops the IPSN method and a robust pilot estimator that outperform existing estimators like POET in heavy-tailed settings.

Findings

The proposed estimator attains sub-Gaussian convergence rates.

It outperforms the POET estimator in heavy-tailed data scenarios.

The method effectively removes the influence of heavy tails on covariance estimation.

Abstract

We study the estimation of high-dimensional covariance matrices under elliptical factor models with 2 + {\epsilon}th moment. For such heavy-tailed data, robust estimators like the Huber-type estimator in Fan, Liu and Wang (2018) can not achieve sub-Gaussian convergence rate. In this paper, we develop an idiosyncratic-projected self-normalization (IPSN) method to remove the effect of heavy-tailed scalar parameter, and propose a robust pilot estimator for the scatter matrix that achieves the sub-Gaussian rate. We further develop an estimator of the covariance matrix and show that it achieves a faster convergence rate than the generic POET estimator in Fan, Liu and Wang (2018).