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Limiting Spectral Distribution of moderately large Kendall's correlation matrix and its application

Raunak Shevade, Monika Bhattacharjee

Abstract

We establish the limiting spectral distribution of Kendall's correlation matrices in the moderate high-dimensional regime where the dimension grows slower than the sample size. Our framework allows observations to be independent but not necessarily identically distributed, and accommodates both discrete and continuous data. Unlike existing results developed under i.i.d. observations, our approach remains valid under substantial distributional heterogeneity and also covers certain i.i.d. models beyond previously studied settings. Under mild symmetry and convergence conditions on some traces, we prove that the empirical spectral distribution of a properly centered and scaled Kendall's correlation matrix converges weakly almost surely to a deterministic, generally model-dependent limit. The analysis clarifies how distributional heterogeneity influences the limiting spectrum. As an application, we propose a graphical tool for detecting dependence among components in high-dimensional data and show that ignoring heterogeneity may lead to spurious detection of dependence.

Limiting Spectral Distribution of moderately large Kendall's correlation matrix and its application

Abstract

We establish the limiting spectral distribution of Kendall's correlation matrices in the moderate high-dimensional regime where the dimension grows slower than the sample size. Our framework allows observations to be independent but not necessarily identically distributed, and accommodates both discrete and continuous data. Unlike existing results developed under i.i.d. observations, our approach remains valid under substantial distributional heterogeneity and also covers certain i.i.d. models beyond previously studied settings. Under mild symmetry and convergence conditions on some traces, we prove that the empirical spectral distribution of a properly centered and scaled Kendall's correlation matrix converges weakly almost surely to a deterministic, generally model-dependent limit. The analysis clarifies how distributional heterogeneity influences the limiting spectrum. As an application, we propose a graphical tool for detecting dependence among components in high-dimensional data and show that ignoring heterogeneity may lead to spurious detection of dependence.
Paper Structure (14 sections, 7 theorems, 72 equations, 1 figure, 4 tables)

This paper contains 14 sections, 7 theorems, 72 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Convergence of empirical spectral distributions
  3. IID observations
  4. Semi-circle LSDs
  5. Examples
  6. Data-driven ad-hoc verification of assumptions via clustering
  7. Test of Independence
  8. Technical details
  9. Proof of Theorem \ref{['thm: A']}
  10. Hoeffding's decomposition of $\mathbb{T}$
  11. Negligibility of $\mathbb{T}_2$
  12. LSD of $\mathbb{T}_1$
  13. Verification of (c1) condition for $\mathbb{T}_1$
  14. Verification of (c2) for $\mathbb{T}_1$

Key Result

Theorem 1

Suppose Assumptions 1, 2, G1, and G2 hold, and $p := p(n) \to \infty$ with $p/n \to 0$ as $n \to \infty$. Then the ESD of $\sqrt{n/p}(\mathbb{T}-D(\mathbb{T}))$ or $2\sqrt{n/p}(G - g_1\mathbf{I}_p)$ converges weakly, almost surely, to a probability distribution whose odd-order moments vanish and who

Figures (1)

  • Figure 1: ECDFs of $\sqrt{n/p}(\mathbb{T}_Z-D(\mathbb{T}_Z))$ (cyan) and $\sqrt{n/p}(\mathbb{T}_X-D(\mathbb{T}_X))$ (red) for Example \ref{['example: 1']} setting; Columns are for $(n,p)=(100,10),(900,30),(1600,40)$ respectively and rows correspond to $\alpha=0,1,2$ respectively.

Theorems & Definitions (25)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Example 1
  • Corollary 2.1
  • Corollary 2.2
  • Remark 4
  • Example 2
  • Remark 5
  • ...and 15 more