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

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.