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Sub-Gaussian High-Dimensional Covariance Matrix Estimation under Elliptical Factor Model with 2 + εth Moment

Yi Ding, Xinghua Zheng

Abstract

We study the estimation of high-dimensional covariance matrices under elliptical factor models with 2 + εth 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).

Sub-Gaussian High-Dimensional Covariance Matrix Estimation under Elliptical Factor Model with 2 + εth Moment

Abstract

We study the estimation of high-dimensional covariance matrices under elliptical factor models with 2 + εth 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).

Paper Structure

This paper contains 26 sections, 13 theorems, 123 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Covariance Matrix Estimation for Heavy-tailed Distributions
  3. Elliptical Factor Models
  4. Our Contributions
  5. Main Results
  6. Settings and Assumptions
  7. Idiosyncratic-Projected Self-Normalized Estimator of Covariance Matrix
  8. Theoretical Properties
  9. Convergence Rates of Idiosyncratic-Projected Self-Normalized Pilot Estimators
  10. Convergence Rates of Generic POET Estimators based on IPSN
  11. Simulation Studies
  12. Simulation Setting
  13. Covariance Matrix Estimators and Evaluation Metrics
  14. Simulation Results
  15. Empirical Studies
  16. ...and 11 more sections

Key Result

Proposition 1

Under Assumptions asump1--assump_factor_structure, if in addition, $p, n\to \infty$ and satisfy $(\log(p))^{2+\gamma}=o(n)$ for some $\gamma>0$ and $\log(n)=O(\log (p))$, then there exists a constant $c_1>0$ such that and

Figures (1)

  • Figure 1: The ratios between $(\widehat{\xi}_t)$ and $(\xi_t)$ from one simulation. We compare our proposed IPSN estimator with the estimator (ZL11) in ZL11. The process $(\xi_t)$ is generated from Pareto distribution $P(\xi_t>x)=(x_m/x)^\alpha$ with the tail parameter $\alpha=2.2$.

Theorems & Definitions (13)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Proposition 2
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 3 more