The state-of-the-art methods for estimating high-dimensional covariance
matrices all shrink the eigenvalues of the sample covariance matrix towards a
data-insensitive shrinkage target. The underlying shrinkage transformation is
either chosen heuristically - without compelling theoretical justification - or
optimally in view of restrictive distributional assumptions. In this paper, we
propose a principled approach to construct covariance estimators without
imposing restrictive assumptions. That is, we study distributionally robust
covariance estimation problems that minimize the worst-case Frobenius error
with respect to all data distributions close to a nominal distribution, where
the proximity of distributions is measured via a divergence on the space of
covariance matrices. We identify mild conditions on this divergence under which
the resulting minimizers represent shrinkage estimators. We show that the
corresponding shrinkage transformations are intimately related to the
geometrical properties of the underlying divergence. We also prove that our
robust estimators are efficiently computable and asymptotically consistent and
that they enjoy finite-sample performance guarantees. We exemplify our general
methodology by synthesizing explicit estimators induced by the
Kullback-Leibler, Fisher-Rao, and Wasserstein divergences. Numerical
experiments based on synthetic and real data show that our robust estimators
are competitive with state-of-the-art estimators.

提出了一种在不强加限制性假设的情况下构建协方差估计器的原则方法，通过最小化与接近名义分布的所有数据分布相关的最坏情况 Frobenius 误差来研究分布鲁棒协方差估计问题，证明了鲁棒估计器的有效计算性、渐近一致性和有限样本性能保证，并通过合成 Kullback-Leibler、Fisher-Rao 和 Wasserstein 散度的显式估计器来说明这一通用方法。基于合成和实际数据的数值实验表明，我们的鲁棒估计器与最先进的估计器具有相竞争的性能。