  We address covariance estimation in the sense of minimum mean-squared error
(MMSE) for Gaussian samples. Specifically, we consider shrinkage methods which
are suitable for high dimensional problems with a small number of samples
(large p small n). First, we improve on the Ledoit-Wolf (LW) method by
conditioning on a sufficient statistic. By the Rao-Blackwell theorem, this
yields a new estimator called RBLW, whose mean-squared error dominates that of
LW for Gaussian variables. Second, to further reduce the estimation error, we
propose an iterative approach which approximates the clairvoyant shrinkage
estimator. Convergence of this iterative method is established and a closed
form expression for the limit is determined, which is referred to as the oracle
approximating shrinkage (OAS) estimator. Both RBLW and OAS estimators have
simple expressions and are easily implemented. Although the two methods are
developed from different persepctives, their structure is identical up to
specified constants. The RBLW estimator provably dominates the LW method.
Numerical simulations demonstrate that the OAS approach can perform even better
than RBLW, especially when n is much less than p. We also demonstrate the
performance of these techniques in the context of adaptive beamforming.
