Two-step estimators of high-dimensional correlation matrices

We investigate block diagonal and hierarchical nested stochastic multivariate Gaussian models by studying their sample cross-correlation matrix on high dimensions. By performing numerical simulations, we compare a filtered sample cross-correlation with the population cross-correlation matrices by using several rotationally invariant estimators (RIEs) and hierarchical clustering estimators (HCEs) under several loss functions. We show that at large but finite sample size, sample cross-correlations filtered by RIE estimators are often outperformed by HCE estimators for several of the loss functions. We also show that for block models and for hierarchically nested block models, the best determination of the filtered sample cross-correlation is achieved by introducing two-step estimators combining state-of-the-art nonlinear shrinkage models with hierarchical clustering estimators.