Test for bandedness of high-dimensional covariance matrices and bandwidth estimation

TL;DR

This paper introduces a new adaptive statistical test to determine if a high-dimensional covariance matrix is banded with diverging bandwidth, along with a consistent estimator for the bandwidth, applicable in 'large p, small n' scenarios.

Contribution

It proposes a novel, distribution-free test for bandedness and a consistent estimator for the bandwidth of high-dimensional covariance matrices, with theoretical and empirical validation.

Findings

The test accurately detects bandedness in simulations.

The bandwidth estimator is consistent and reliable.

Empirical analysis demonstrates practical applicability.

Abstract

Motivated by the latest effort to employ banded matrices to estimate a high-dimensional covariance $\Sigma$, we propose a test for $\Sigma$ being banded with possible diverging bandwidth. The test is adaptive to the "large $p$, small $n$" situations without assuming a specific parametric distribution for the data. We also formulate a consistent estimator for the bandwidth of a banded high-dimensional covariance matrix. The properties of the test and the bandwidth estimator are investigated by theoretical evaluations and simulation studies, as well as an empirical analysis on a protein mass spectroscopy data.