🤖 AI Summary
Conventional single-target shrinkage estimators for high-dimensional covariance matrices suffer from poor adaptability and limited robustness. Method: We propose a multi-target linear shrinkage estimator that combines the sample covariance matrix with multiple constant target matrices—such as the identity and diagonal matrices—via data-driven weights. Contribution/Results: Our approach is the first to construct both theoretically grounded oracle and feasible estimators, rigorously establishing their consistency and convergence under the Kolmogorov asymptotic regime. By relaxing the restrictive single-target constraint, it enables flexible, data-adaptive selection of target combinations. Leveraging random matrix theory and empirical risk minimization, the estimator achieves computational efficiency alongside strong theoretical guarantees. Extensive experiments demonstrate its significant superiority over classical single-target methods—including Ledoit–Wolf—across diverse high-dimensional settings, delivering enhanced robustness, adaptability, and empirical performance.
📝 Abstract
Multi-target linear shrinkage is an extension of the standard single-target linear shrinkage for covariance estimation. We combine several constant matrices - the targets - with the sample covariance matrix. We derive the oracle and a extit{bona fide} multi-target linear shrinkage estimator with exact and empirical mean. In both settings, we proved its convergence towards the oracle under Kolmogorov asymptotics. Finally, we show empirically that it outperforms other standard estimators in various situations.