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).

翻译：在本文中,我们研究对缺失数据的共变估算。 我们考虑的是与数据独立的缺失数据机制,或者有时间差异的依赖性。 此外, 观察的变量可能具有任意( 非统一) 和依附性观察概率。 对于每个机制,我们构建一个公正的估算器,并获得其估计误差在操作器规范中的预期值的界限。 我们的界限相当于, 最高为恒定和对数因素, 相当于完整和统一缺失观测的艺术界限。 此外, 对于更普遍的非统一和依附性案例, 拟议的界限是新的, 或比以前的结果有所改进。 我们的错误估计值取决于我们所要求的大小有效等级, 即对缺失的观察进行总体估计。 所有在这项工作中研究的估算师都具有相同的同步率( 逻辑系数 ) 。