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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 variables is larger than the number of observations. Shrinkage approaches for estimating a high-dimensional covariance matrix are often 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 matrix 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 covariance 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 covariance 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.