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This article proposes a method to consistently estimate functionals$\frac1p\sum_{i=1}^pf(\lambda_i(C_1C_2))$ of the eigenvalues of the product oftwo covariance matrices $C_1,C_2\in\mathbb{R}^{p\times p}$ based on theempirical estimates $\lambda_i(\hat C_1\hat C_2)$ ($\hatC_a=\frac1{n_a}\sum_{i=1}^{n_a} x_i^{(a)}x_i^{(a){{\sf T}}}$), when the size$p$ and number $n_a$ of the (zero mean) samples $x_i^{(a)}$ are similar. As acorollary, a consistent estimate of the Wasserstein distance (related to thecase $f(t)=\sqrt{t}$) between centered Gaussian distributions is derived.  The new estimate is shown to largely outperform the classical samplecovariance-based `plug-in' estimator. Based on this finding, a practicalapplication to covariance estimation is then devised which demonstratespotentially significant performance gains with respect to state-of-the-artalternatives.

Random Matrix-Improved Estimation of the Wasserstein Distance between two Centered Gaussian Distributions