Linear Pooling of Sample Covariance Matrices
Linear Pooling of Sample Covariance Matrices
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DOI：
10.1109/tsp.2021.3139207
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发表时间：
2020-08
影响因子：
5.4
通讯作者：
Elias Raninen;David E. Tyler;E. Ollila
中科院分区：
文献类型：
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作者：
Elias Raninen;David E. Tyler;E. Ollila
我们考虑在样本大小与数据维度相当的情况下估计 $K$ 群体或类别的高维协方差矩阵的问题。我们建议将每个类协方差矩阵估计为所有类样本协方差矩阵的不同线性组合。当样本大小有限时，这种方法可以减少估计误差，并且真实的类协方差矩阵具有某种程度上相似的结构。我们开发了一种有效的方法来估计线性组合中的系数，该方法在样本是从具有有限四阶矩的（未指定）椭圆对称分布中抽取的一般假设下最小化均方误差。为此，我们利用空间符号协方差矩阵，我们（在相当一般的条件下）证明它是随着维度增长到无穷大而归一化协方差矩阵的渐近无偏估计量。我们还展示了如何使用所提出的方法在单类协方差矩阵估计问题中为多个目标矩阵选择正则化参数。我们通过数值模拟研究评估所提出的方法，包括使用真实股票数据在全局最小方差投资组合优化中的应用。
We consider the problem of estimating high-dimensional covariance matrices of $K$-populations or classes in the setting where the sample sizes are comparable to the data dimension. We propose estimating each class covariance matrix as a distinct linear combination of all class sample covariance matrices. This approach is shown to reduce the estimation error when the sample sizes are limited, and the true class covariance matrices share a somewhat similar structure. We develop an effective method for estimating the coefficients in the linear combination that minimize the mean squared error under the general assumption that the samples are drawn from (unspecified) elliptically symmetric distributions possessing finite fourth-order moments. To this end, we utilize the spatial sign covariance matrix, which we show (under rather general conditions) to be an asymptotically unbiased estimator of the normalized covariance matrix as the dimension grows to infinity. We also show how the proposed method can be used in choosing the regularization parameters for multiple target matrices in a single class covariance matrix estimation problem. We assess the proposed method via numerical simulation studies including an application in global minimum variance portfolio optimization using real stock data.