Minimax optimal estimation of general bandable covariance matrices

Cai et al. (2010)  [4] have studied the minimax optimal estimation of a collection of large bandable covariance matrices whose off-diagonal entries decay to zero at a polynomial rate. They have shown that the minimax optimal procedures are fundamentally different under Frobenius and spectral norms, regardless of the rate of polynomial decay. To gain more insight into this interesting problem, we study minimax estimation of large bandable covariance matrices over a parameter space characterized by a general positive decay function. We obtain explicit results to show how the decay function determines the minimax rates of convergence and the optimal procedures. From the general minimax analysis we find that for certain decay functions there is a tapering estimator that simultaneously attains the minimax optimal rates of convergence under the two norms. Moreover, we show that under the ultra-high dimension scenario it is possible to achieve adaptive minimax optimal estimation under the spectral norm. These new findings complement previous work.