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

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.

Published in at http://dx.doi.org/10.1214/12-AOS1002 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Regularized estimation of large covariance matrices
• Covariance regularization by thresholding
• Optimal rates of convergence for covariance matrix estimation
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• The asymptotic distribution and Berry--Esseen bound of a new test for independence in high dimension with an application to stochastic optimization

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