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Restricted Block Permutation for Two-Sample Testing

Jungwoo Ho

TL;DR

<p>This work introduces a block-restricted one-swap permutation framework for two-sample testing that preserves exact finite-sample validity while concentrating permutation changes along high-contrast, block-restricted paths. By modeling the permutation trajectory as a bounded martingale difference sequence and applying Bernstein–Freedman concentration, the authors derive data-dependent tail bounds and show that increment variances scale as $O(h^2)$ for key statistics, a contraction relative to full relabeling. This variance contraction translates into substantially smaller permutation critical values and improved power for canonical statistics such as the difference in means and the unbiased $\widehat{MMD}^2$, with explicit formulas for the data-dependent power and critical values. The paper also provides practical design guidelines (block formation, complementary block–pair swaps, and choice of representative ratio $\rho$) and supports the theory with simulations demonstrating higher power while maintaining exact type-I error control.</p>

Abstract

We study a structured permutation scheme for two-sample testing that restricts permutations to single cross-swaps between block-selected representatives. Our analysis yields three main results. First, we provide an exact validity construction that applies to any fixed restricted permutation set. Second, for both the difference of sample means and the unbiased $\widehat{\mathrm{MMD}}^{2}$ estimator, we derive closed-form one-swap increment identities whose conditional variances scale as $O(h^{2})$, in contrast to the $Θ(h)$ increment variability under full relabeling. This increment-level variance contraction sharpens the Bernstein--Freedman variance proxy and leads to substantially smaller permutation critical values. Third, we obtain explicit, data-dependent expressions for the resulting critical values and statistical power. Together, these results show that block-restricted one-swap permutations can achieve strictly higher power than classical full permutation tests while maintaining exact finite-sample validity, without relying on pessimistic worst-case Lipschitz bounds.

Restricted Block Permutation for Two-Sample Testing

TL;DR

<p>This work introduces a block-restricted one-swap permutation framework for two-sample testing that preserves exact finite-sample validity while concentrating permutation changes along high-contrast, block-restricted paths. By modeling the permutation trajectory as a bounded martingale difference sequence and applying Bernstein–Freedman concentration, the authors derive data-dependent tail bounds and show that increment variances scale as for key statistics, a contraction relative to full relabeling. This variance contraction translates into substantially smaller permutation critical values and improved power for canonical statistics such as the difference in means and the unbiased , with explicit formulas for the data-dependent power and critical values. The paper also provides practical design guidelines (block formation, complementary block–pair swaps, and choice of representative ratio ) and supports the theory with simulations demonstrating higher power while maintaining exact type-I error control.</p>

Abstract

We study a structured permutation scheme for two-sample testing that restricts permutations to single cross-swaps between block-selected representatives. Our analysis yields three main results. First, we provide an exact validity construction that applies to any fixed restricted permutation set. Second, for both the difference of sample means and the unbiased estimator, we derive closed-form one-swap increment identities whose conditional variances scale as , in contrast to the increment variability under full relabeling. This increment-level variance contraction sharpens the Bernstein--Freedman variance proxy and leads to substantially smaller permutation critical values. Third, we obtain explicit, data-dependent expressions for the resulting critical values and statistical power. Together, these results show that block-restricted one-swap permutations can achieve strictly higher power than classical full permutation tests while maintaining exact finite-sample validity, without relying on pessimistic worst-case Lipschitz bounds.

Paper Structure

This paper contains 29 sections, 6 theorems, 63 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Our perspective.
  3. Contributions.
  4. Setup and Block--Restricted One-Swap Construction
  5. Two-sample framework and notation
  6. Block partition and representative ratio
  7. Validity for arbitrary restricted permutation sets
  8. Concentration Along Restricted Swap Paths
  9. Transposition distance and admissible paths
  10. Bernstein--Freedman bound for permutation increments
  11. Variance Identities for Canonical Two-Sample Statistics
  12. Difference in sample means
  13. Unbiased MMD
  14. Summary: variance contraction
  15. From Variance to Critical Values and Power
  16. ...and 14 more sections

Key Result

Theorem 2.1

arbitrary Let $S \subseteq S_N$ be any fixed subset of permutations. Sample $\sigma_0, \sigma_1, \dots, \sigma_M \stackrel{iid}{\sim} \mathrm{Unif}(S)$, and define Then $P$ is a valid $p$-value in the sense that, under $H_0$,

Figures (1)

  • Figure 1: Simulation results of the complementary block--pair permutation design. Left: power improvement relative to classical full relabeling. Right: type-I error control at $\alpha=0.05$. The external legend summarizes color/line mapping.

Theorems & Definitions (10)

  • Theorem 2.1: Validity under arbitrary restricted permutations
  • Corollary 2.2: Validity of the block--restricted one-swap scheme
  • Definition 3.1: Transposition distance and admissible path
  • Theorem 3.2: Bernstein--Freedman bound for transposition increments
  • proof
  • Lemma 4.1: One-swap update and exact variance
  • proof
  • Lemma 4.2: Exact one-swap decomposition
  • proof
  • Theorem 4.3: Variance contraction under block--restricted one-swaps