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Stratified Permutational Berry--Esseen Bounds and Their Applications to Statistics

Pengfei Tian, Fan Yang, Peng Ding

Abstract

The stratified linear permutation statistic arises in various statistics problems, including stratified and post-stratified survey sampling, stratified and post-stratified experiments, conditional permutation tests, etc. Although we can derive the Berry--Esseen bounds for the stratified linear permutation statistic based on existing bounds for the non-stratified statistics, those bounds are not sharp, and moreover, this strategy does not work in general settings with heterogeneous strata with varying sizes. We first use Stein's method to obtain a unified stratified permutational Berry--Esseen bound that can accommodate heterogeneous strata. We then apply the bound to various statistics problems, leading to stronger theoretical quantifications and thereby facilitating statistical inference in those problems.

Stratified Permutational Berry--Esseen Bounds and Their Applications to Statistics

Abstract

The stratified linear permutation statistic arises in various statistics problems, including stratified and post-stratified survey sampling, stratified and post-stratified experiments, conditional permutation tests, etc. Although we can derive the Berry--Esseen bounds for the stratified linear permutation statistic based on existing bounds for the non-stratified statistics, those bounds are not sharp, and moreover, this strategy does not work in general settings with heterogeneous strata with varying sizes. We first use Stein's method to obtain a unified stratified permutational Berry--Esseen bound that can accommodate heterogeneous strata. We then apply the bound to various statistics problems, leading to stronger theoretical quantifications and thereby facilitating statistical inference in those problems.

Paper Structure

This paper contains 106 sections, 45 theorems, 346 equations, 1 table.

Table of Contents

  1. Introduction to Stratified Permutational Berry--Esseen Bounds
  2. Stratified Linear Permutation Statistic and Overview of the Main Result
  3. Motivating Examples
  4. Literature Review, Open Questions, and Our Contributions
  5. Basic result and notation
  6. Suboptimal Results Based on Existing BEBs
  7. Leverage the BEB for the Sum of Independent Random Variables
  8. Leverage the BEB for the Non-stratified Linear Permutation Statistic
  9. Stratified Permutational BEBs Based on Stein's Method
  10. Main Theorem
  11. Corollaries
  12. Sketch of the Proof and Additional Results
  13. Review of Stein's Method
  14. Simplification
  15. Zero-Bias Transformation
  16. ...and 91 more sections

Key Result

Theorem 1

Standardize the stratified linear permutation statistic $W_{A,\pi}$ defined in eq:matrix permutation to have mean $0$ and variance $1$. There exists a universal constant $C$, such that

Theorems & Definitions (55)

  • Theorem 1: informal version
  • Example 1: Stratified sampling
  • Example 2: Stratified experiment
  • Example 3: Post-stratified sampling
  • Example 4: Post-stratified experiment
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Proposition 3
  • Corollary 2
  • ...and 45 more