Double shrinkage correction in sample LMMSE estimation

The sample linear minimum mean square error (LMMSE) estimator undergoes high performance degradation in the small sample size regime. Herein a double shrinkage correction is proposed to alleviate this problem. First, an affine transformation of the sample covariance matrix (SCM) is considered within the LMMSE. Second, a linear transformation of that modified filter is proposed. The linear transformation minimizes the asymptotic MSE of the filter given a shrinkage of the SCM. And the shrinkage of the SCM optimizes the asymptotic MSE of the data covariance. Simulations highlight that the proposed estimator outperforms robust methods to the small sample size, namely LMMSE based on diagonal loading (DL) or Ledoit-Wolf (LW) regularizations of the SCM.