Large covariance matrix estimation: Bridging shrinkage and tapering approaches

In this paper, we propose a shrinkage-to-tapering oracle (STO) estimator for estimation of large covariance matrix when the number of samples is substantially fewer than the number of variables, by combining the strength from both Steinian-type shrinkage and tapering estimators. Our contributions include: (i) Deriving the Frobenius risk and a lower bound for the spectral risk of an MMSE shrinkage estimator; (ii) Deriving a closed-form expression for the optimal coefficient of the proposed STO estimator. Simulations on auto-regression (e.g. a sparse case) and fraction Brownian motion (e.g. a non-sparse case) covariance structures are used to demonstrate the superiority of the proposed estimator.