  High-dimensional statistical tests often ignore correlations to gain
simplicity and stability leading to null distributions that depend on
functionals of correlation matrices such as their Frobenius norm and other
$\ell_r$ norms. Motivated by the computation of critical values of such tests,
we investigate the difficulty of estimation the functionals of sparse
correlation matrices. Specifically, we show that simple plug-in procedures
based on thresholded estimators of correlation matrices are sparsity-adaptive
and minimax optimal over a large class of correlation matrices. Akin to
previous results on functional estimation, the minimax rates exhibit an elbow
phenomenon. Our results are further illustrated in simulated data as well as an
empirical study of data arising in financial econometrics.
