Consistent Estimation of a Class of Distances Between Covariance Matrices
This work addresses the need for accurate distance estimation between covariance matrices in multivariate analytical contexts, offering a robust statistical framework, though it appears incremental as it builds on existing distance families.
The paper tackles the problem of estimating distances between covariance matrices directly from data, focusing on a family of distances expressed as sums of traces of functions applied to each matrix. The result includes a consistent estimator that outperforms conventional plug-in methods in empirical evaluations, along with a central limit theorem providing asymptotic Gaussianity and closed-form expressions for means and variances.
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.