Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution

TL;DR

This paper provides a theoretical analysis of Kendall's tau-based correlation matrix estimators for transelliptical distributions, demonstrating their convergence properties without requiring moment conditions, and highlighting the role of effective rank.

Contribution

It introduces a comprehensive theoretical framework for Kendall's tau estimators in transelliptical models, including convergence rates and the sign sub-Gaussian condition, advancing robust high-dimensional correlation estimation.

Findings

Kendall's tau estimator converges under spectral and restricted spectral norms.

Effective rank influences the convergence rate of the estimator.

The sign sub-Gaussian condition ensures fast convergence without moment assumptions.

Abstract

Correlation matrices play a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, it is not an effective estimator when facing heavy-tailed distributions. As a robust alternative, Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of "effective rank" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a "sign sub-Gaussian condition" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the fast rate of convergence. In both cases, we do not need any moment condition.