Recent advances in shrinkage-based high-dimensional inference

Recently, the shrinkage approach has increased its popularity in theoretical and applied statistics, especially, when point estimators for high-dimensional quantities have to be constructed. A shrinkage estimator is usually obtained by shrinking the sample estimator towards a deterministic target. This allows to reduce the high volatility that is commonly present in the sample estimator by introducing a bias such that the mean-square error of the shrinkage estimator becomes smaller than the one of the corresponding sample estimator. The procedure has shown great advantages especially in the high-dimensional problems where, in general case, the sample estimators are not consistent without imposing structural assumptions on model parameters. In this paper, we review the mostly used shrinkage estimators for the mean vector, covariance and precision matrices. The application in portfolio theory is provided where the weights of optimal portfolios are usually determined as functions of the mean vector and covariance matrix. Furthermore, a test theory on the mean–variance optimality of a given portfolio based on the shrinkage approach is presented as well.