Abstract: For large covariance matrices and the corresponding precision matrices with banding structures, this paper develops a criterion to identify the bandwidth. The new method is based on an objective function that is discontinuous at the true bandwidth to show a "valley-cliff" pattern so that the identification of this location can be visualized and easily implemented. We offer the estimation consistency and the estimation error bound of the estimated covariance matrix and precision matrix with this estimated bandwidth. Numerical studies demonstrate the finite sample validity of the method, and a real data validity analysis is used for illustration.

Key words and phrases: Banding, Cholesky decomposition, covariance matrix, large p small n, precision matrix, tapering.