Consistent Estimation of a Class of Distances Between Covariance Matrices

TL;DR

This paper introduces a consistent method for estimating a broad class of distances between covariance matrices directly from data, accounting for the Riemannian geometry of positive definite matrices, with proven asymptotic properties and empirical validation.

Contribution

It proposes a novel consistent estimator for covariance matrix distances within a Riemannian framework, along with a central limit theorem for statistical inference.

Findings

Estimator outperforms conventional plug-in methods

Central limit theorem provides asymptotic Gaussianity and variance estimates

Empirical results confirm improved accuracy in multivariate analysis

Abstract

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.