In this paper, we consider the problem of testing equality of the covariance matrices of LL complex Gaussian multivariate time series of dimension MM. We study the special case where each of the LL covariance matrices is modeled as a rank KK perturbation of the identity matrix, corresponding to a signal plus noise model. A new test statistic based on the estimates of the eigenvalues of the different covariance matrices is proposed. In particular, we show that this statistic is consistent and with controlled type I error in the high-dimensional asymptotic regime where the sample sizes N1,…,NLN_{1},\dots,N_{L} of each time series and the dimension MM both converge to infinity at the same rate, while KK and LL are kept fixed. We also provide some simulations on synthetic and real data (SAR images) which demonstrate significant improvements over some classical methods such as the GLRT, or other alternative methods relevant for the high-dimensional regime and the low-rank model.

中文翻译：

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检验高维高斯低秩模型协方差相等性的新统计量


在本文中，我们考虑检验MM维的LL复高斯多元时间序列的协方差矩阵的相等性的问题。我们研究特殊情况，其中每个 LL 协方差矩阵都被建模为单位矩阵的秩 KK 扰动，对应于信号加噪声模型。提出了一种基于不同协方差矩阵特征值估计的新检验统计量。特别是，我们表明该统计量在高维渐进状态下是一致的并且具有受控的 I 类误差，其中每个时间序列的样本大小 N1,…,NLN_{1},\dots,N_{L} 和维度MM 都以相同的速率收敛到无穷大，而 KK 和 LL 保持固定。我们还提供了一些对合成数据和真实数据（SAR 图像）的模拟，这些模拟证明了相对于一些经典方法（例如 GLRT）或与高维体系和低秩模型相关的其他替代方法的显着改进。