Nonparametric Stein-type Shrinkage Covariance Matrix Estimators in High-Dimensional Settings

Estimating a covariance matrix is an important task in applications where the number of vari-ables is larger than the number of observations. In the literature, shrinkage approaches for estimat-ing a high-dimensional covariance matrix are employed to circumvent the limitations of the sample covariance matrix. A new family of nonparametric Stein-type shrinkage covariance estimators is proposed whose members are written as a convex linear combination of the sample covariance ma-trix and of a predefined invertible target matrix. Under the Frobenius norm criterion, the optimal shrinkage intensity that defines the best convex linear combination depends on the unobserved co-variance matrix and it must be estimated from the data. A simple but effective estimation process that produces nonparametric and consistent estimators of the optimal shrinkage intensity for three popular target matrices is introduced. In simulations, the proposed Stein-type shrinkage covari-ance matrix estimator based on a scaled identity matrix appeared to be up to 80 % more efficient than existing ones in extreme high-dimensional settings. A colon cancer dataset was analyzed to demonstrate the utility of the proposed estimators. A rule of thumb for adhoc selection among the three commonly used target matrices is recommended