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Decoupling and randomization for double-indexed permutation statistics

Mingxuan Zou, Jingfan Xu, Peng Ding, Fang Han

TL;DR

The paper addresses concentration for double-indexed permutation statistics (DIPS) by decomposing $Q_w$ into a single-indexed component and a degenerate DIPS $Q_d$, and then developing new decoupling and randomization inequalities to bound MGFs. It yields a genuine combinatorial Hanson-Wright-type inequality and a combinatorial Bennett inequality, with dimension-free tails that depend on norms like $\|\mathbf{C}\|_{F}$, $\|\mathbf{C}\|_{op}$, and related Hadamard-structured terms. The framework is illustrated through nonparametric examples (Mann-Whitney–Wilcoxon, Daniels, Chatterjee, Friedman–Rafsky) and an application to causal inference via regression adjustment under complete randomization. Together, these results provide new tools for concentration in design-based inference and broaden the scope of permutation-statistic analysis in statistics and econometrics.

Abstract

This paper introduces a version of decoupling and randomization to establish concentration inequalities for double-indexed permutation statistics. The results yield, among other applications, a new combinatorial Hanson-Wright inequality and a new combinatorial Bennett inequality. Several illustrative examples from rank-based statistics, graph-based statistics, and causal inference are also provided.

Decoupling and randomization for double-indexed permutation statistics

TL;DR

The paper addresses concentration for double-indexed permutation statistics (DIPS) by decomposing into a single-indexed component and a degenerate DIPS , and then developing new decoupling and randomization inequalities to bound MGFs. It yields a genuine combinatorial Hanson-Wright-type inequality and a combinatorial Bennett inequality, with dimension-free tails that depend on norms like , , and related Hadamard-structured terms. The framework is illustrated through nonparametric examples (Mann-Whitney–Wilcoxon, Daniels, Chatterjee, Friedman–Rafsky) and an application to causal inference via regression adjustment under complete randomization. Together, these results provide new tools for concentration in design-based inference and broaden the scope of permutation-statistic analysis in statistics and econometrics.

Abstract

This paper introduces a version of decoupling and randomization to establish concentration inequalities for double-indexed permutation statistics. The results yield, among other applications, a new combinatorial Hanson-Wright inequality and a new combinatorial Bennett inequality. Several illustrative examples from rank-based statistics, graph-based statistics, and causal inference are also provided.
Paper Structure (24 sections, 15 theorems, 225 equations)

This paper contains 24 sections, 15 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Generalized combinatorial quadratic forms
  4. Examples in nonparametric statistics
  5. An example in causal inference
  6. Main results
  7. Proofs of the main results
  8. Auxiliary lemmas
  9. Proof of Theorem \ref{['thm:decoupling']}
  10. Proof of Theorem \ref{['thm:randomization']}
  11. Proof of Theorem \ref{['thm:main']}
  12. Proof of Corollary \ref{['cor:uncentered2']}
  13. Proof of Theorem \ref{['thm:HansonWright']}
  14. Proof of Theorem \ref{['thm:Bennett']}
  15. Proofs of auxiliary lemmas
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $N\geq 20$. Assume that $\mathbf{C}$ is doubly centered and $\mathbf{A} =\{a_{ij}\}_{i,j\in[N]}$ has entries in $[-1,1]$. There then exists a universal constant $K>0$ such that the following hold true for any $t>0$.

Theorems & Definitions (34)

  • Theorem 1.1: A combinatorial Hanson-Wright-type inequality
  • Theorem 1.2: A combinatorial Bennett-type inequality
  • Example 1.1: Mann-Whitney-Wilcoxon statistic
  • Example 1.2: Daniels’ generalized correlation coefficient
  • Example 1.3: Chatterjee's rank correlation
  • Example 1.4: Friedman and Rafsky's graph correlation
  • Theorem 2.1: Combinatorial decoupling
  • Theorem 2.2: Combinatorial randomization
  • Theorem 2.3
  • Corollary 2.1
  • ...and 24 more