This work considers the problem of estimating the distance between two covariance matrices directly from the data. Particularly, we are interested in the family of distances that can be expressed as sums of traces of functions that are separately applied to each covariance matrix. This family of distances is particularly useful as it takes into consideration the fact that covariance matrices lie in the Riemannian manifold of positive definite matrices, thereby including a variety of commonly used metrics, such as the Euclidean distance, Jeffreys' divergence, and the log-Euclidean distance. Moreover, a statistical analysis of the asymptotic behavior of this class of distance estimators has also been conducted. Specifically, we present a central limit theorem that establishes the asymptotic Gaussianity of these estimators and provides closed form expressions for the corresponding means and variances. Empirical evaluations demonstrate the superiority of our proposed consistent estimator over conventional plug-in estimators in multivariate analytical contexts. Additionally, the central limit theorem derived in this study provides a robust statistical framework to assess of accuracy of these estimators.

翻译：本研究探讨直接从数据估计两个协方差矩阵间距离的问题。特别地，我们关注可表示为分别作用于各协方差矩阵的函数迹之和的距离族。该距离族具有特殊价值，因为它考虑了协方差矩阵位于正定矩阵黎曼流形这一几何特性，从而涵盖了多种常用度量，如欧氏距离、Jeffreys散度及对数欧氏距离。此外，本文还对这类距离估计量的渐近性质进行了统计分析。具体而言，我们提出了一个中心极限定理，确立了这些估计量的渐近高斯性，并给出了相应均值与方差的闭合表达式。实证评估表明，在多变量分析场景中，我们提出的一致估计量优于传统的插件估计量。同时，本研究推导的中心极限定理为评估这些估计量的精度提供了稳健的统计框架。