# Powerful rank verification for multivariate Gaussian data with any covariance structure

arXiv: https://arxiv.org/abs/2503.01065 · HTML: https://mlsanity.com/paper/2503.01065/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 22 references that are in the map.
- First posted on arXiv 3 Mar 2025.

## Builds on (most cited first, 3 of 3)

- [Exact post-selection inference, with application to the lasso](https://mlsanity.com/paper/1311.6238/) (2013)
- [Rank Verification for Exponential Families](https://mlsanity.com/paper/1610.03944/) (2016)
- [Selective inference is easier with p-values](https://mlsanity.com/paper/2411.13764/) (2024)

## Built on by (most cited first, 2 of 2)

- [Locally minimax optimal confidence sets for the best model](https://mlsanity.com/paper/2503.21639/) (2025)
- [How Sensitive Are LLM Leaderboard Claims to Hidden Model Selection?](https://mlsanity.com/paper/2609.28177/) (2026)

## Abstract (arXiv)

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.