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

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