Multi-Target Shrinkage Estimation for Covariance Matrices

Covariance matrix estimation is problematic when the number of samples is relatively small compared with the number of variables. One way to tackle this problem is through the use of shrinkage estimators that offer a compromise between the sample covariance matrix and a well-conditioned matrix (also known as the ”target”) with the aim of minimizing the mean-squared error (MSE). The use of only one target limits the shrinkage estimators´ flexibility when minimizing the MSE. In this paper, we propose a multi-target shrinkage estimator (MTSE) for covariance matrices that exploits the Lediot-Wolf (LW) method by utilizing several targets simultaneously. This greatly increases the estimator´s flexibility and enables it to attain a lower MSE. We also offer a general target that serves as a framework for designing a wide variety of targets. In consequence, instead of studying individual targets, the general framework can be utilized. We then show that the framework encompasses several targets that already exist in the literature. Numerical simulations demonstrate that the MTSE significantly reduces the MSE and is highly effective in classification tasks.