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

Let $X_1,...,X_n\in\mathbb{R}^{p}$ be a sample from an elliptical distribution with correlation matrix $\rho$ and Kendall's tau correlation matrix $\tau$. Besides the minimax rate $c_{n,p}(\frac{\log p}{n})^{1-q/2}$ of estimation for $\rho$ under the Frobenius norm over large classes of correlation matrices with sparse rows, where the parameters $c_{n,p}$ and $q$ depend on the class of sparse correlation matrices, we establish a critical threshold level regarding the minimax rate for a natural threshold estimator based on Kendall's tau sample correlation matrix $\hat\tau$. More precisely we identify a constant $\alpha^\ast>0$ such that the proposed estimator attains the minimax rate for any entrywise threshold level $\alpha(\frac{\log p}{n})^{1/2}$ with $\alpha>\alpha^\ast$. In general this is not anymore true for $\alpha<\alpha^\ast$. This critical value $\alpha^\ast$ is given by $\frac{\sqrt{2}\pi}{3}$ and therefore by choosing $\alpha$ slightly larger than $\alpha^\ast$ the corresponding estimator does not only achieve the minimax rate but provides a non-trivial estimate of the true correlation matrix even for moderate sample sizes $n$. The main ingredient to provide the critical threshold level is a sharp large deviation result for Kendall's tau sample correlation if the underlying $2$-dimensional elliptical distribution implies weak correlation between the components. This result is evolved from an asymptotic expansion of the number of permutations with a certain number of inversions. To the best of the authors knowledge this is the first work concerning critical threshold constants.