Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model

Abstract

Covariance matrix estimation is one of the most important problems in statistics. To accommodate the complexity of modern datasets, it is desired to have estimation procedures that not only can incorporate the structural assumptions of covariance matrices, but are also robust to outliers from arbitrary sources. In this paper, we define a new concept called matrix depth and then propose a robust covariance matrix estimator by maximizing the empirical depth function. The proposed estimator is shown to achieve minimax optimal rate under Huber's $\epsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

References in corpus (5)

• Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution
• High Dimensional Robust M-Estimation: Asymptotic Variance via Approximate Message Passing
• Multivariate Analysis of Nonparametric Estimates of Large Correlation Matrices
• Variance Breakdown of Huber (M)-estimators: $n/p\to m\in (1,\infty )$
• Volume Ratio, Sparsity, and Minimaxity under Unitarily Invariant Norms

Cited by in corpus (6)

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• Estimation of the covariance structure of heavy-tailed distributions
• Computationally Efficient Robust Estimation of Sparse Functionals
• Robust Regression via Mutivariate Regression Depth
• Convex programming approach to robust estimation of a multivariate Gaussian model
• Robust Estimation of Covariance Matrices: Adversarial Contamination and Beyond