Powerful rank verification for multivariate Gaussian data with any covariance structure
This addresses a statistical inference problem for researchers analyzing multivariate data, offering a more powerful method than current approaches, though it builds incrementally on prior selective inference work.
The paper tackles the problem of verifying whether the largest K observations in multivariate Gaussian data correspond to the largest K means, generalizing a prior method to any K and covariance structure. It shows that their procedure can draw correct inferences even when a standard test fails, and argues it makes existing simultaneous inference approaches inadmissible for n > 2.
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.