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Robust covariance and scatter matrix estimation under Huber’s contamination model. (English) Zbl 1408.62104
Summary: 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 P. J. Huber’s [Ann. Math. Stat. 35, 73–101 (1964; Zbl 0136.39805); ibid. 36, 1753–1758 (1965; Zbl 0137.12702)] \(\varepsilon\)-contamination model for estimating covariance/scatter matrices with various structures including bandedness and sparsity.

MSC:
62H12 Estimation in multivariate analysis
62C20 Minimax procedures in statistical decision theory
62F35 Robustness and adaptive procedures (parametric inference)
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