Ties, Tails and Spectra: On Rank-Based Dependency Measures in High Dimensions

TL;DR

This paper investigates the spectral distribution of rank-based dependency measures like Kendall's tau and Spearman's rho in high-dimensional settings, extending results to discrete, heavy-tailed data with ties.

Contribution

It provides distribution-free asymptotic results for these measures in high dimensions, including adjustments for ties and heavy tails, filling a gap in existing literature.

Findings

Distribution-free limiting spectral distributions derived for Kendall's tau and Spearman's rho.

Adjustment needed for Kendall's tau to handle tied data.

Limiting eigenvalue distribution results for random matrices with rows on the Euclidean sphere.

Abstract

This work is concerned with the limiting spectral distribution of rank-based dependency measures in high dimensions. We provide distribution-free results for multivariate empirical versions of Kendall's $\tau$ and Spearman's $\rho$ in a setting where the dimension $p$ grows at most proportionally to the sample size $n$. Although rank-based measures are known to be well suited for discrete and heavy-tailed data, previous works in the field focused mostly on the continuous and light-tailed case. We close this gap by imposing mild assumptions and allowing for general types of distributions. Interestingly, our analysis reveals that a non-trivial adjustment of classical Kendall's $\tau$ is needed to obtain a pivotal limiting distribution in the presence of tied data. The proof for Spearman's $\rho$ is facilitated by a result regarding the limiting eigenvalue distribution of a general class of random matrices with rows on the Euclidean unit sphere, which is of independent interest. For instance, this finding can be used to derive the limiting spectral distribution of sample correlation matrices, which, in contrast to most existing works, accommodates heavy-tailed data.