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.Comment: To appear in Computational Statistics and Data Analysi
