Cram\'er-type moderate deviation for double index permutation statistics

TL;DR

This paper proves a Cramér-type moderate deviation theorem for double-index permutation statistics, improving upon previous bounds and applicable to both classical and sparse permutation-based statistics.

Contribution

It introduces a new moderate deviation result for DIPS that achieves optimal convergence rates and extends to sparse statistics, surpassing prior Berry-Esseen bounds.

Findings

Recover the optimal convergence rates for classical DIPS.

Extend moderate deviation results to sparse permutation statistics.

Require more easily verifiable conditions for the theorem.

Abstract

We establish a Cram\'er-type moderate deviation theorem for double-index permutation statistics (DIPS). To the best of our knowledge, previous results only provided Berry-Esseen type bounds for DIPS, which cannot yield moderate deviation results and are insufficient to capture the optimal convergence rates for some relatively sparse DIPS. Our result overcome these limitations: it not only recover the optimal convergence rates for classical DIPS, such as the Mann-Whitney-Wilcoxon statistic, but also extend to sparse statistics, including the number of descents in permutations and Chatterjee's rank correlation coefficient, for which previous approaches do not apply. To prove this result, we establish a Cram\'er-type moderate deviation of normal approximation for bounded exchangeable pairs. Compared with existing results, our theorem requires more easily verifiable conditions.