High-dimensional covariance matrix estimation has become a hot topic. I have two theoretic papers and one method and theory paper in this area:
- In the to-appar-in-Bernoulli paper On the sample covariance matrix estimator of reduced effective rank population matrices, with applications to fPCA, I studied theoretical properties of the sample covariance matrix as an estimator of population matrices of reduced effective rank. The effective rank of a matrix is the ratio of its trace to its largest singular value, and provides a measure of complexity.
- In the in-revision paper On the theoretic and practical merits of the banding estimator for large covariance matrices, I proved that the banding estimator proposed by Bickel and Levina achieves rate-optimality under the operator norm, for a class of approximately banded covariance matrices. In addition, I proposed a Stein’s unbiased risk estimate (Sure)-type approach for selecting the bandwidth for the banding estimator.
- In the under-review paper Convex banding of covariance matrices, a new sparse estimator of high-dimensional bandable covariance matrices was proposed. The estimator, which is the solution to a convex optimization problem, is equivalently expressed as an estimator which tapers the sample covariance matrix by a Toeplitz, sparsely-banded, data-adaptive matrix. As a result of this adaptivity, the convex banding estimator enjoys theoretical optimality properties not attained by previous banding or tapered estimators.