The Annals of Statistics

Robust covariance and scatter matrix estimation under Huber’s contamination model

Mengjie Chen, Chao Gao, and Zhao Ren

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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 $\varepsilon$-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 1932-1960.

Received: March 2016
Revised: June 2017
First available in Project Euclid: 17 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62C20: Minimax procedures

Data depth Minimax rate high-dimensional statistics outliers contamination model breakdown point


Chen, Mengjie; Gao, Chao; Ren, Zhao. Robust covariance and scatter matrix estimation under Huber’s contamination model. Ann. Statist. 46 (2018), no. 5, 1932--1960. doi:10.1214/17-AOS1607.

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Supplemental materials

  • Supplement to “Robust covariance and scatter matrix estimation under Huber’s contamination model”. In this supplement, we collect the proofs for the remaining main results, provide details on the extension to the noncentered observations and demonstrate numerical studies in low-to-moderate dimensional settings.