Powerful rank verification for multivariate Gaussian data with any covariance structure

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, citeGutmann 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.