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Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$

Ruoyu Wu

TL;DR

This work characterizes the limiting spectral distribution of the high-dimensional multivariate Kendall–τ matrix. Using a bridge to covariance-type matrices and Stieltjes-transform techniques, it proves that the ESD of $\tfrac{1}{2}pK_n$ converges to the Marčenko–Pastur law with variance $\tfrac{1}{2}$ when $p/n \to y \in (0,1)$ in the i.i.d. setting, and extends to a general independent-component model via a fixed-point equation for the LSD of $\tfrac{1}{2}\mathrm{tr}\Sigma K_n$. The results are corroborated by comprehensive simulations across distributions and aspect ratios, illustrating the robustness and universality of the spectral behavior. The findings bolster robust high-dimensional inference by placing Kendall–τ based matrices in the same universality class as covariance-type estimators, with explicit LSD characterizations under affine data transformations. Potential applications include robust PCA, dependence estimation, and factor-analysis workflows, while future work may address edge eigenvalue behavior, convergence rates, and extensions to broader data structures.

Abstract

The multivariate Kendall-$τ$ statistic, denoted by $K_n$, plays a significant role in robust statistical analysis. This paper establishes the limiting properties of the empirical spectral distribution (ESD) of $K_n$. We demonstrate that the ESD of $\frac{1}{2}pK_n$ converges almost surely to the Marčenko--Pastur law with variance parameter $\frac{1}{2}$, analogous to the classical result for sample covariance matrices. Using Stieltjes transform techniques, we extend these results to the independent component model, deriving a fixed-point equation that characterizes the limiting spectral distribution of $\frac{1}{2}trΣK_n$. The theoretical findings are validated through comprehensive simulation studies.

Limiting Spectral Distribution of High-dimensional Multivariate Kendall-$τ$

TL;DR

This work characterizes the limiting spectral distribution of the high-dimensional multivariate Kendall–τ matrix. Using a bridge to covariance-type matrices and Stieltjes-transform techniques, it proves that the ESD of converges to the Marčenko–Pastur law with variance when in the i.i.d. setting, and extends to a general independent-component model via a fixed-point equation for the LSD of . The results are corroborated by comprehensive simulations across distributions and aspect ratios, illustrating the robustness and universality of the spectral behavior. The findings bolster robust high-dimensional inference by placing Kendall–τ based matrices in the same universality class as covariance-type estimators, with explicit LSD characterizations under affine data transformations. Potential applications include robust PCA, dependence estimation, and factor-analysis workflows, while future work may address edge eigenvalue behavior, convergence rates, and extensions to broader data structures.

Abstract

The multivariate Kendall- statistic, denoted by , plays a significant role in robust statistical analysis. This paper establishes the limiting properties of the empirical spectral distribution (ESD) of . We demonstrate that the ESD of converges almost surely to the Marčenko--Pastur law with variance parameter , analogous to the classical result for sample covariance matrices. Using Stieltjes transform techniques, we extend these results to the independent component model, deriving a fixed-point equation that characterizes the limiting spectral distribution of . The theoretical findings are validated through comprehensive simulation studies.
Paper Structure (15 sections, 11 theorems, 113 equations, 4 figures, 1 table)

This paper contains 15 sections, 11 theorems, 113 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Notation and Preliminaries
  4. Main Results
  5. Proofs
  6. Auxiliary Lemmas
  7. Proofs of the Main Theorems
  8. Proof of Theorem \ref{['thm2.1']}
  9. Proof of Theorem \ref{['thm2.2']}
  10. Relationship Between the Two Theorems
  11. Simulation Studies
  12. Simulation Design
  13. Applications
  14. Applications in Statistics
  15. Future Research Directions

Key Result

Theorem 2.1

Let $\{X_{ij}\}$$(i=1,\ldots,p; j=1,\ldots,n)$ be independent and identically distributed random variables with mean $0$, variance $1$, and finite fourth moment. Define $X_i = (X_{1i}, X_{2i}, \ldots, X_{pi})^T$. If $p/n \to y \in (0,1)$, then the empirical spectral distribution of $\tfrac{1}{2}pK_n where $a = \tfrac{1}{2}(1 - \sqrt{y})^2$ and $b = \tfrac{1}{2}(1 + \sqrt{y})^2$.

Figures (4)

  • Figure 1: Empirical spectral distribution of $\frac{1}{2}pK_n$ under $N(0,1)$ data.
  • Figure 2: Empirical spectral distribution under $N(0,2)$ data.
  • Figure 3: Empirical spectral distribution under $U(0,1)$ data.
  • Figure 4: Empirical spectral distribution under $U(0,2)$ data.

Theorems & Definitions (15)

  • Theorem 2.1: The limiting spectral distribution in the i.i.d. case
  • Remark 2.1
  • Theorem 2.2: General model with independent components
  • Lemma 3.1: Pairwise Convergence Lemma
  • proof
  • Lemma 3.2: Maximal Partial Sum Lemma
  • proof
  • Lemma 3.3: Estimation of Lévy distance
  • Lemma 3.4
  • Lemma 3.5
  • ...and 5 more