Tests of missing completely at random based on sample covariance matrices
基于样本协方差矩阵的完全随机缺失检验AI 解读 ↓
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基于样本协方差矩阵的完全随机缺失检验
- 研究问题
- 在维度可能随样本量增长的数据中，如何利用不同缺失模式下协方差矩阵的兼容性，检测完全随机缺失假设的违背？
- 方法与思路
- 作者将问题放宽为协方差兼容性检验，用半定规划刻画相关矩阵的不兼容度，通过代入估计量和自助法构造检验。
- 材料陈述的贡献
- 摘要称，论文给出关键对偶结果，在某些缺失结构下达到极小极大速率最优，并展示对不兼容分布的检验功效与矩阵补全用途。
- 阅读关注点
- 核对完全随机缺失与协方差兼容性的逻辑关系、维度及尾部条件，以及自助校准和极小极大最优结论的具体范围。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；兼容性检验是对原问题的放宽，未拒绝不能据此确认完全随机缺失，重尾证据来自模拟。
原文摘要
We study the problem of testing whether the missing values of a potentially high-dimensional dataset are Missing Completely at Random (MCAR). We relax the problem of testing MCAR to the problem of testing the compatibility of a collection of covariance matrices, motivated by the fact that this procedure is feasible when the dimension grows with the sample size. Our first contributions are to define a natural measure of the incompatibility of a collection of correlation matrices, which can be characterised as the optimal value of a Semidefinite Programming (SDP) problem, and to establish a key duality result allowing its practical computation and interpretation. By analysing the concentration properties of the natural plug-in estimator for this measure, we propose a novel hypothesis test, which is calibrated via a bootstrap procedure and demonstrates power against any distribution with incompatible covariance matrices. By considering key examples of missingness structures, we demonstrate that our procedures are minimax rate optimal in certain cases. We further validate our methodology with numerical simulations that provide evidence of validity and power, even when data are heavy tailed. Furthermore, tests of compatibility can be used to test the feasibility of positive semidefinite matrix completion problems with noisy observations, and thus our results may be of independent interest.