提出可处理任意协方差结构的高维正态数据秩验证方法
Powerful rank verification for multivariate Gaussian data with any covariance structure
- 基于选择性推断，推广了原方法至任意K值和任意协方差结构
- 当最小标准化差异检验拒绝时，可准确推断前K大观测来自前K大均值
- 适用于高维独立或等相关数据，且优于传统同时推断方法
在观测n维多元正态数据时，何时能确信最大的K个观测值来自最大的K个均值？当K=1且协方差为各向同性时，‘Gutmann’认为仅当双侧均值差检验拒绝原假设时此推断成立。本文利用选择性推断工具，将该方法推广至任意K值和任意协方差结构。我们证明：当最小标准化差异的双侧均值差检验拒绝时，该程序可作出正确推断，且在某些检验未拒绝的情况下仍有效。基于此，我们论证当n > 2时，现有同时推断方法不再最优。在观测独立（方差可能不等）或等相关情形下，该程序等价于对位于前K与外部的观测中最小标准化差异对执行双侧均值差检验。
原文摘要 · Abstract (English)
Upon observing $n$-dimensional multivariate Gaussian data, when can we infer that the largest $K$ observations came from the largest $K$ means? When $K=1$ and the covariance is isotropic, \cite{Gutmann} argue that this inference is justified when the two-sided difference-of-means test comparing the largest and second largest observation rejects. Leveraging tools from selective inference, we provide a generalization of their procedure that applies for both any $K$ and any covariance structure. We show that our procedure draws the desired inference whenever the two-sided difference-of-means test comparing the pair of observations inside and outside the top $K$ with the smallest standardized difference rejects, and sometimes even when this test fails to reject. Using this insight, we argue that our procedure renders existing simultaneous inference approaches inadmissible when $n > 2$. When the observations are independent (with possibly unequal variances) or equicorrelated, our procedure corresponds exactly to running the two-sided difference-of-means test comparing the pair of observations inside and outside the top $K$ with the smallest standardized difference.
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