Shrinkage estimators for large covariance matrices in multivariate real and complex normal distributions under an invariant quadratic loss

The problem of estimating large covariance matrices of multivariate real normal and complex normal distributions is considered when the dimension of the variables is larger than the number of sample size. The Stein-Haff identities and calculus on eigenstructures for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain class of estimators for the population covariance matrices under an invariant quadratic loss functions are obtained for real and complex cases, respectively. Based on the unbiased risk estimates, shrinkage estimators which are counterparts of the estimators due to Haff [1980, Ann. Statist. 8 ∗ 2000 Mathematics Subject Classification. Primary: 62H12, Secondary: 62F10.