Abstract: Estimations of high-dimensional banded covariance matrices are widely used in multivariate statistical analysis. To ensure the validity of such estimations, we test the hypothesis that the covariance matrix is banded with a certain bandwidth under a high-dimensional framework. Though several testing methods have been proposed in the literature, these tests are only powerful for some alternatives with certain sparsity levels, but not others. Here, we propose two adaptive tests for the bandedness of a high-dimensional covariance matrix that is powerful for alternatives with various sparsity levels. The proposed methods can also be used to test the banded structure of the covariance matrices of the error vectors in high-dimensional factor models. Based on these statistics, we introduce a consistent bandwidth estimator for a banded high-dimensional covariance matrix. We use simulation studies and an application to a prostate cancer data set to evaluate the effectiveness of the proposed adaptive tests and bandwidth estimator.

Key words and phrases: Asymptotic normality, banded covariance matrix, high-dimensional hypothesis testing, sparsity level, U-statistics.