Optimal Estimation for a Class of High‐Dimensional Covariance Matrices

Estimation of large covariance matrices plays important roles in high‐dimensional data analysis. The sub‐Gaussian condition of a random vector is usually assumed, which requires the existence of infinite moments. Avella‐Medina et al. (2018) provide an upper bound estimation in the sense of probability over a sparse covariance matrix space under the weak assumption of bounded moments (), see Biometrika, 105, 271–284. In particular, their estimation attains the minimax optimality when . The authors conjecture that their estimation is optimal as well for . In this paper, we first extend their upper bound estimation to a larger space and then prove the optimality of our estimation. This can be considered as a solution to their conjecture. Moreover, we give an optimal estimation in terms of expectation on the same space. Finally, numerical experiments support our theoretical analysis.