In this paper, we consider the problem of testing equality of the covariance matrices of L complex Gaussian multivariate time series of dimension $M$ . We study the special case where each of the L covariance matrices is modeled as a rank K 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 $N_1,\ldots,N_L$ of each time series and the dimension $M$ both converge to infinity at the same rate, while $K$ and $L$ 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.

翻译：本文研究$L$个$M$维复高斯多元时间序列协方差矩阵相等性检验问题。我们重点探讨每种协方差矩阵均建模为单位矩阵的$K$秩扰动（对应信号加噪声模型）的特殊情形。基于不同协方差矩阵特征值的估计，提出了一种新的检验统计量。特别地，我们证明在高维渐近框架下（各时间序列样本量$N_1,\ldots,N_L$与维度$M$以相同速率趋于无穷，而$K$和$L$保持固定），该统计量具有一致性和可控的第一类错误率。通过合成数据与真实数据（SAR图像）的仿真实验，我们证实该方法相较经典方法（如GLRT）及其他适用于高维场景与低秩模型的替代方法具有显著性能提升。