  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.
