Journal Article

User-Friendly Covariance Estimation for Heavy-Tailed Distributions

Ke, Yuan, Minsker, Stanislav, Zhou, Wen-Xin, Ren, Zhao, Sun, Qiang

2019

Overview

We provide a survey of recent results on covariance estimation for heavy-tailed distributions. By unifying ideas scattered in the literature, we propose user-friendly methods that facilitate practical implementation. Specifically, we introduce elementwise and spectrumwise truncation operators, as well as their 𝛭-estimator counterparts, to robustify the sample covariance matrix. Different from the classical notion of robustness that is characterized by the breakdown property, we focus on the tail robustness which is evidenced by the connection between nonasymptotic deviation and confidence level. The key insight is that estimators should adapt to the sample size, dimensionality and noise level to achieve optimal tradeoff between bias and robustness. Furthermore, to facilitate practical implementation, we propose data-driven procedures that automatically calibrate the tuning parameters. We demonstrate their applications to a series of structured models in high dimensions, including the bandable and low-rank covariance matrices and sparse precision matrices. Numerical studies lend strong support to the proposed methods.

• Publisher: Institute of Mathematical Statistics
• Subject: Confidence intervals / Covariance matrix / Estimating techniques / Noise levels / Numerical analysis / Operators (mathematics) / Robustness (mathematics) / Sparsity