"A Revisit to Estimation of the Precision Matrix of the Wishart Distribution"

The estimation of the precision matrix of the Wishart distribution is one of classical problems studied in a decision-theoretic framework and is related to estimation of mean and covariance matrices of a multivariate normal distribution. This paper revisits the estimation problem of the precision matrix and investigates how it connects with the theory of the covariance estimation from a decision-theoretic aspect. To evaluate estimators in terms of risk functions, we employ two kinds of loss functions: the non-scale-invariant loss and the scale-invariant loss functions which are induced from estimation of means. Using the same methods as in the estimation of the covariance matrix, we derive not only the James-Stein type of estimators improving on the Stein type one under the non-scale-invariant loss. It is observed that dominance properties given in the estimation of the covariance matrix do not necessarily hold in our setup under the non-scale-invariant loss, but still hold relative to the scale-invariant loss. The simulation studies are given, and estimators having superior risk performances are proposed.