In this article, we first propose generalized row/column matrix Kendall's tau for matrix-variate observations that are ubiquitous in areas such as finance and medical imaging. For a random matrix following a matrix-variate elliptically contoured distribution, we show that the eigenspaces of the proposed row/column matrix Kendall's tau coincide with those of the row/column scatter matrix respectively, with the same descending order of the eigenvalues. We perform eigenvalue decomposition to the generalized row/column matrix Kendall's tau for recovering the loading spaces of the matrix factor model. We also propose to estimate the pair of the factor numbers by exploiting the eigenvalue-ratios of the row/column matrix Kendall's tau. Theoretically, we derive the convergence rates of the estimators for loading spaces, factor scores and common components, and prove the consistency of the estimators for the factor numbers without any moment constraints on the idiosyncratic errors. Thorough simulation studies are conducted to show the higher degree of robustness of the proposed estimators over the existing ones. Analysis of a financial dataset of asset returns and a medical imaging dataset associated with COVID-19 illustrate the empirical usefulness of the proposed method.

中文翻译：

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矩阵 Kendall 的高维 tau：矩阵因子模型的稳健统计

在本文中，我们首先提出广义行/列矩阵 Kendall 的 tau，用于在金融和医学成像等领域普遍存在的矩阵变量观察。对于遵循矩阵变量椭圆轮廓分布的随机矩阵，我们表明所提出的行/列矩阵 Kendall's tau 的特征空间分别与行/列散布矩阵的特征空间一致，特征值的降序相同。我们对广义行/列矩阵 Kendall 的 tau 执行特征值分解，以恢复矩阵因子模型的加载空间。我们还建议通过利用行/列矩阵 Kendall 的 tau 的特征值比率来估计因子数对。理论上，我们推导出加载空间的估计器的收敛速度，因子分数和公共组件，并证明因子数估计量的一致性，而对特殊误差没有任何矩约束。进行了彻底的模拟研究，以显示所提出的估计器比现有估计器具有更高程度的鲁棒性。对资产回报的金融数据集和与 COVID-19 相关的医学成像数据集的分析说明了所提出方法的经验有用性。