In this paper we study covariance estimation with missing data. We consider missing data mechanisms that can be independent of the data, or have a time varying dependency. Additionally, observed variables may have arbitrary (non uniform) and dependent observation probabilities. For each mechanism, we construct an unbiased estimator and obtain bounds for the expected value of their estimation error in operator norm. Our bounds are equivalent, up to constant and logarithmic factors, to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent cases, the proposed bounds are new or improve upon previous results. Our error estimates depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. All the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).

中文翻译：

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具有非均匀和数据相关缺失观测值的协方差矩阵估计

在本文中，我们研究了缺失数据的协方差估计。我们考虑可以独立于数据或具有随时间变化的依赖性的缺失数据机制。此外，观察到的变量可能具有任意（非均匀）和相关的观察概率。对于每种机制，我们构建一个无偏估计器并获得它们在算子范数中的估计误差的期望值的界限。我们的界限等价于常数和对数因子，相当于完整和统一缺失观察的最新界限。此外，对于更一般的非均匀和依赖情况，建议的界限是新的或在先前结果的基础上有所改进。我们的误差估计取决于我们称之为缩放有效秩的数量，它概括了有效秩以解释缺失的观察结果。