The critical threshold level on Kendall's tau statistic concerning minimax estimation of sparse correlation matrices

TL;DR

This paper identifies the critical threshold level for Kendall's tau-based estimator to achieve minimax optimality in estimating sparse correlation matrices in high-dimensional elliptical models, providing new theoretical insights.

Contribution

It introduces the first critical threshold constant for Kendall's tau estimators, enabling optimal calibration without tail information and extending to transelliptical models.

Findings

Establishes the critical threshold constant st for Gaussian models.

Achieves minimax rates under Frobenius and spectral norms.

Provides a sharp large deviation expansion for Kendall's tau.

Abstract

In a sparse high-dimensional elliptical model we consider a hard threshold estimator for the correlation matrix based on Kendall's tau with threshold level $\alpha(\frac{\log p}{n})^{1/2}$. Parameters $\alpha$ are identified such that the threshold estimator achieves the minimax rate under the squared Frobenius norm and the squared spectral norm. This allows a reasonable calibration of the estimator without any quantitative information about the tails of the underlying distribution. For Gaussian observations we even establish a critical threshold constant $\alpha^\ast$ under the squared Frobenius norm, i.e. the proposed estimator attains the minimax rate for $\alpha>\alpha^\ast$ but in general not for $\alpha<\alpha^\ast$. To the best of the author's knowledge this is the first work concerning critical threshold constants. The main ingredient to provide the critical threshold level is a sharp large deviation expansion for Kendall's tau sample correlation evolved from an asymptotic expansion of the number of permutations with a certain number of inversions. The investigation of this paper also covers further statistical problems like the estimation of the latent correlation matrix in the transelliptical and nonparanormal family.