This work addresses large dimensional covariance matrix estimation with unknown mean. The empirical covariance estimator fails when dimension and number of samples are proportional and tend to infinity, settings known as Kolmogorov asymptotics. When the mean is known, Ledoit and Wolf (2004) proposed a linear shrinkage estimator and proved its convergence under those asymptotics. To the best of our knowledge, no formal proof has been proposed when the mean is unknown. To address this issue, we propose a new estimator and prove its quadratic convergence under the Ledoit and Wolf assumptions. Finally, we show empirically that it outperforms other standard estimators.

翻译：本文研究了未知均值条件下的大维协方差矩阵估计问题。当维度与样本数量成比例且趋近于无穷大时（即Kolmogorov渐近框架），经验协方差估计失效。在已知均值的情况下，Ledoit与Wolf（2004）提出了线性收缩估计量，并证明了其在该渐近条件下的收敛性。然而据我们所知，当均值未知时，尚无正式的理论证明。针对这一问题，本文提出了一种新型估计量，并在Ledoit-Wolf假设条件下证明了其二次收敛性。最后，实证结果表明该估计量优于其他标准估计方法。