The critical 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 $\tau$ such that the distributions of the components $X_{i1}$, $i=1,...,p$, have no atoms. Then $\sin(\frac{\pi}{2}\hat\tau_{ij})$ is a well-behaved estimator for the entry $\rho_{ij}$, where $\hat\tau_{ij}$ is Kendall's tau sample correlation based on $(X_{i1},X_{j1}),...,(X_{in},X_{jn})$. We study the family of entrywise threshold estimators $\{\hat\rho_\alpha|\alpha>0\}$, where $\hat\rho_\alpha=:\hat\rho=(\hat\rho_{ij})$ consists of the entries $$\hat\rho_{ij}:=\sin\left(\frac{\pi}{2}\hat\tau_{ij}\right)\mathrm{1}\left\{\left|\sin\left(\frac{\pi}{2}\hat\tau_{ij}\right)\right|>\alpha\sqrt{\frac{\log p}{n}}\right\} \text{ for }i\neq j\text{ and }\hat\rho_{ii}=1.$$ In particular, we raise the question how large the threshold constant $\alpha$ needs to be so that $\hat\rho$ attains the minimax rate under the Frobenius norm over all permissible elliptical distributions, which suffice a sparsity condition on the rows of the correlation matrix $\rho$. It is shown that $\hat\rho$ achieves the optimal rate $c_{n,p}(\frac{\log p}{n})^{1-q/2}$ if $\alpha>\pi$, where the parameters $c_{n,p}$ and $q$ depend on the class of sparse correlation matrices. For Gaussian observations we even establish a critical threshold constant, i.e. we identify a constant $\alpha^\ast>0$ such that the proposed estimator attains the minimax rate for $\alpha>\alpha^\ast$ but in general not for $\alpha<\alpha^\ast$. This critical value $\alpha^\ast$ is given by $\frac{\sqrt{2}\pi}{3}$.