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 (RIE) and hierarchical clustering estimators (HCE) under several loss functions. We show that at large but finite sample size, sample cross-correlation 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 non-linear shrinkage models with hierarchical clustering estimators.

翻译：本文研究了对角块结构和分层嵌套结构的高维多元高斯随机模型，通过分析高维情况下的样本互相关矩阵。我们进行数值模拟，在多种损失函数下，利用若干旋转不变估计器（RIE）和层次聚类估计器（HCE）对滤波后的样本互相关矩阵与总体互相关矩阵进行比较。结果表明，在有限大样本量下，对于多种损失函数，经RIE滤波的样本互相关矩阵往往不如HCE估计器表现优异。我们还发现，对于块模型和分层嵌套块模型，通过引入结合最先进非线性收缩模型与层次聚类估计器的两步估计方法，能实现滤波样本互相关矩阵的最优确定。