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We derive an optimal shrinkage sample covariance matrix (SCM) estimator whichis suitable for high dimensional problems and when sampling from an unspecifiedelliptically symmetric distribution. Specifically, we derive the optimal(oracle) shrinkage parameters that obtain the minimum mean-squared error (MMSE)between the shrinkage SCM and the true covariance matrix when sampling from anelliptical distribution. Subsequently, we show how the oracle shrinkageparameters can be consistently estimated under the random matrix theory regime.Simulations show the advantage of the proposed estimator over the conventionalshrinkage SCM estimator due to Ledoit and Wolf (2004). The proposed shrinkageSCM estimator often provides significantly better performance than theLedoit-Wolf estimator and has the advantage that consistency is guaranteed overthe whole class of elliptical distributions with finite 4th order moments.

Optimal High-Dimensional Shrinkage Covariance Estimation for Elliptical Distributions