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.
"Test for bandedness of high-dimensional covariance matrices and bandwidth estimation." Ann. Statist. 40 (3) 1285 - 1314, June 2012. https://doi.org/10.1214/12-AOS1002