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Cramér-type moderate deviation for double index permutation statistics

Songhao Liu, Qiman Shao, Jingyu Xu

Abstract

We establish a Cramér-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 Cramér-type moderate deviation of normal approximation for bounded exchangeable pairs. Compared with existing results, our theorem requires more easily verifiable conditions.

Cramér-type moderate deviation for double index permutation statistics

Abstract

We establish a Cramér-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 Cramér-type moderate deviation of normal approximation for bounded exchangeable pairs. Compared with existing results, our theorem requires more easily verifiable conditions.

Paper Structure

This paper contains 15 sections, 11 theorems, 226 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Main Results
  3. Applications
  4. The Chatterjee's rank correlation coefficient
  5. The Number of descents and inversions of permutation
  6. The Mann-Whitney-Wilcoxon statistic
  7. Proof of main results
  8. A decomposition of general DIPS
  9. Cramér-type moderate deviation for bounded exchangeable pairs
  10. Proof of Theorem \ref{['thm-general-double-index-permutation-statistics-a-b-form']}
  11. Proof of Theorem \ref{['thm-general-MD']}
  12. Proof of other results
  13. Proof of Theorem \ref{['theo-chatterjee-rank-correlation-coefficient']}
  14. Proof of Theorem \ref{['theo-descents-inversions']}
  15. Proof of Theorem \ref{['theo-mann-whitney-wilcoxon']}

Key Result

Proposition 2.1

By a suitable normalization, the general DIPS (general_DIPS) can be converted into one of the following forms where $\{a(i,k)\}_{i,k\in[N]}$ is a real number matrix and $\{b(i,j,k,l)\}_{i,j,k,l\in[N]}$ is a 4-index real number array, and $\{a(i,k)\}_{i,k\in[N]}$ satisfies and and $\{b(i,j,k,l)\}_{i,j,k,l\in[N]}$ satisfies no matter what $\{a(i,k)\}_{i,k\in[N]}$ is.

Theorems & Definitions (26)

  • Proposition 2.1
  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Theorem 3.1
  • Remark 3
  • Theorem 3.2
  • Theorem 3.3
  • proof : Proof of Proposition \ref{['general-DIPS-convert']}
  • Theorem 4.1
  • ...and 16 more