We construct an estimator $\widehat{\Sigma}$ for covariance matrices of unknown, centred random vectors X, with the given data consisting of N independent measurements $X_1,...,X_N$ of X and the wanted confidence level. We show under minimal assumptions on X, the estimator performs with the optimal accuracy with respect to the operator norm. In addition, the estimator is also optimal with respect to direction dependence accuracy: $\langle \widehat{\Sigma}u,u\rangle$ is an optimal estimator for $\sigma^2(u)=\mathbb{E}\langle X,u\rangle^2$ when $\sigma^2(u)$ is ``large".

翻译：我们针对未知、中心化的随机向量X的协方差矩阵构造了一个估计量$\widehat{\Sigma}$，已知数据由X的N次独立测量$X_1,...,X_N$以及所需的置信水平构成。我们证明，在关于X的极弱假设条件下，该估计量在算子范数意义下达到了最优精度。此外，该估计量在方向依赖精度上也是最优的：当$\sigma^2(u)=\mathbb{E}\langle X,u\rangle^2$“较大”时，$\langle \widehat{\Sigma}u,u\rangle$是$\sigma^2(u)$的最优估计量。