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Some Asymptotic Results on Multiple Testing under Weak Dependence

Swarnadeep Datta, Monitirtha Dey

TL;DR

The asymptotic behaviors of the classical Bonferroni and the Sidak procedure are explored; and it is shown that both of these control FWER at the desired level exactly as the number of hypotheses approaches infinity.

Abstract

This paper studies the means-testing problem under weakly correlated Normal setups. Although quite common in genomic applications, test procedures having exact FWER control under such dependence structures are nonexistent. We explore the asymptotic behaviors of the classical Bonferroni (when adjusted suitably) and the Sidak procedure; and show that both of these control FWER at the desired level exactly as the number of hypotheses approaches infinity. We derive analogous limiting results on the generalized family-wise error rate and power. Simulation studies depict the asymptotic exactness of the procedures empirically.

Some Asymptotic Results on Multiple Testing under Weak Dependence

TL;DR

The asymptotic behaviors of the classical Bonferroni and the Sidak procedure are explored; and it is shown that both of these control FWER at the desired level exactly as the number of hypotheses approaches infinity.

Abstract

This paper studies the means-testing problem under weakly correlated Normal setups. Although quite common in genomic applications, test procedures having exact FWER control under such dependence structures are nonexistent. We explore the asymptotic behaviors of the classical Bonferroni (when adjusted suitably) and the Sidak procedure; and show that both of these control FWER at the desired level exactly as the number of hypotheses approaches infinity. We derive analogous limiting results on the generalized family-wise error rate and power. Simulation studies depict the asymptotic exactness of the procedures empirically.
Paper Structure (24 sections, 11 theorems, 64 equations, 8 tables)

This paper contains 24 sections, 11 theorems, 64 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Power Analysis
  5. Extension to Both-sided Testing
  6. Simulation Studies
  7. Simulation Scheme
  8. Simulation Results
  9. Adjusted Bonferroni (One-sided)
  10. Sidak (One-sided)
  11. Adjusted Bonferroni (Two-sided)
  12. Sidak (Two-sided)
  13. Concluding Remarks
  14. Proofs of theoretical results mentioned in Section 3
  15. Proof of \ref{['probabilistic_k_dn']}
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $\left\{X_n\right\}$ be a weakly dependent standard Gaussian sequence. Suppose $\left\{u_n\right\}$ be a sequence and $\left\{d_n\right\}$ be a positive integer sequence of n such that $d_n\to\infty$ and ${d_n}(1-\Phi(u_n))\to \tau\text{ as }n\to\infty$ for some $\tau \in [0,\,\infty]$. For $1 \

Theorems & Definitions (16)

  • Theorem 3.1
  • Theorem 3.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 4.1
  • Proposition 1
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • ...and 6 more