Adaptive Tests for Bandedness of High-dimensional Covariance Matrices

TL;DR

This paper introduces an adaptive testing method for high-dimensional banded covariance matrices that is effective across various sparsity levels, along with a consistent bandwidth estimator, validated through simulations and real data.

Contribution

It proposes a new adaptive test for bandedness in high-dimensional covariance matrices that works across different sparsity structures, plus a consistent bandwidth estimator.

Findings

The new test outperforms existing methods in simulations.

The bandwidth estimator is consistent and reliable.

Application to real data demonstrates practical usefulness.

Abstract

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.