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Maximum of sparsely equicorrelated Gaussian fields and applications

Johannes Heiny, Tiefeng Jiang, Tuan Pham, Yongcheng Qi

Abstract

We investigate the extreme values of a sparse and equicorrelated Gaussian field on a triangle: the correlations on every vertical or horizontal line are all equal to a parameter $r \in [0,1/2]$ and are zero everywhere else. This problem is closely linked with various problems in high-dimensional statistics and extreme-value theory. We identify the threshold for $r$ at which the standard Gumbel law breaks down. Our result is based on a subtle application of the Chen-Stein method for Poisson approximation. As applications, we discuss the implication of our results on multiple testing and resolve several questions that were left open in \cite{heiny2024maximum}, \cite{tang2022asymptotic} and \cite{Jiang19}.

Maximum of sparsely equicorrelated Gaussian fields and applications

Abstract

We investigate the extreme values of a sparse and equicorrelated Gaussian field on a triangle: the correlations on every vertical or horizontal line are all equal to a parameter and are zero everywhere else. This problem is closely linked with various problems in high-dimensional statistics and extreme-value theory. We identify the threshold for at which the standard Gumbel law breaks down. Our result is based on a subtle application of the Chen-Stein method for Poisson approximation. As applications, we discuss the implication of our results on multiple testing and resolve several questions that were left open in \cite{heiny2024maximum}, \cite{tang2022asymptotic} and \cite{Jiang19}.
Paper Structure (13 sections, 12 theorems, 156 equations)

This paper contains 13 sections, 12 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Application I: Maximum interpoint distance
  4. The fixed marginals case
  5. The $n$-dependent marginals case
  6. Application II: Sample coefficients of equicorrelated populations
  7. The fixed marginal case
  8. The $n$-dependent marginals case
  9. Application III: FWER control in mutiple testing
  10. Proof of Theorem \ref{['infty']}
  11. Proof of Theorem \ref{['critical']}
  12. Proof of Theorem \ref{['0']}
  13. Technical results

Key Result

Theorem 2.1

Suppose $(1-2r) \frac{\sqrt{\log n}}{\log \log n} \to \infty$ and recall the Gaussian field $\mathcal{G}_n$ in sparse equi GF. Then, converges in distribution to the standard Gumbel law with CDF $\exp \left( - e^{-x} \right)$.

Theorems & Definitions (15)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.1
  • Proposition 3.1
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.2
  • Proposition 4.1
  • Proposition 4.2
  • ...and 5 more