Robust Covariance Matrix Estimation via Matrix Depth
Robust Covariance Matrix Estimation via Matrix Depth
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发表时间：
2015-06
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通讯作者：
Mengjie Chen;Chao Gao;Zhao Ren
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作者：
Mengjie Chen;Chao Gao;Zhao Ren
协方差矩阵估计是统计学中最重要的问题之一。为了适应现代数据集的复杂性，需要有一种估计程序，不仅可以包含协方差矩阵的结构假设，而且对于任意来源的异常值也具有鲁棒性。在本文中，我们定义了一个称为矩阵深度的新概念，并通过最大化经验深度函数提出了一种鲁棒的协方差矩阵估计器。所提出的估计器在 Huber 的 $\epsilon$-污染模型下实现了极小最大最优率，用于估计具有包括带状和稀疏性在内的各种结构的协方差/散布矩阵。
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 we 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.