This paper considers the regularized estimation of covariance matrices (CM) of high-dimensional (compound) Gaussian data for minimum variance distortionless response (MVDR) beamforming. Linear shrinkage is applied to improve the accuracy and condition number of the CM estimate for low-sample-support cases. We focus on data-driven techniques that automatically choose the linear shrinkage factors for shrinkage sample covariance matrix ($text{S}^2$CM) and shrinkage Tylers estimator (STE) by exploiting cross validation (CV). We propose leave-one-out cross-validation (LOOCV) choices for the shrinkage factors to optimize the beamforming performance, referred to as $text{S}^2$CM-CV and STE-CV. The (weighted) out-of-sample output power of the beamfomer is chosen as a proxy of the beamformer performance and concise expressions of the LOOCV cost function are derived to allow fast optimization. For the large system regime, asymptotic approximations of the LOOCV cost functions are derived, yielding the $text{S}^2$CM-AE and STE-AE. In general, the proposed algorithms are able to achieve near-oracle performance in choosing the linear shrinkage factors for MVDR beamforming. Simulation results are provided for validating the proposed methods.
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