Multi-level Thresholding Test for High Dimensional Covariance Matrices

TL;DR

This paper introduces a multi-level thresholding test for high-dimensional covariance matrices that effectively detects sparse and faint differences, achieving optimal detection rates across various regimes.

Contribution

It develops a novel multi-thresholding test with a new U-statistic composition, improving detection of sparse, weak covariance differences in high dimensions.

Findings

The test is powerful in detecting sparse and faint differences.

It attains the optimal minimax detection boundary.

Simulation studies confirm its practical utility.

Abstract

We consider testing the equality of two high-dimensional covariance matrices by carrying out a multi-level thresholding procedure, which is designed to detect sparse and faint differences between the covariances. A novel U-statistic composition is developed to establish the asymptotic distribution of the thresholding statistics in conjunction with the matrix blocking and the coupling techniques. We propose a multi-thresholding test that is shown to be powerful in detecting sparse and weak differences between two covariance matrices. The test is shown to have attractive detection boundary and to attain the optimal minimax rate in the signal strength under different regimes of high dimensionality and the sparsity of the signal. Simulation studies are conducted to demonstrate the utility of the proposed test.