Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix

Abstract

Many applications require an estimate for the covariance matrix that is non-singular and well-conditioned. As the dimensionality increases, the sample covariance matrix becomes ill-conditioned or even singular. A common approach to estimating the covariance matrix when the dimensionality is large is that of Stein-type shrinkage estimation. A convex combination of the sample covariance matrix and a well-conditioned target matrix is used to estimate the covariance matrix. Recent work in the literature has shown that an optimal combination exists under mean-squared loss, however it must be estimated from the data. In this paper, we introduce a new set of estimators for the optimal convex combination for three commonly used target matrices. A simulation study shows an improvement over those in the literature in cases of extreme high-dimensionality of the data. A data analysis shows the estimators are effective in a discriminant and classification analysis.