ABSTRACT 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.
Li et al. (Wed,) studied this question.