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

Motivated by the latest effort to employ banded matrices to estimate a high-dimensional covariance $Σ$, we propose a test for $Σ$ 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.

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