Estimating covariance matrices II
Journal of Multivariate Analysis
Fundamentals of statistical signal processing: estimation theory
Fundamentals of statistical signal processing: estimation theory
Expectations of useful complex Wishart forms
Multidimensional Systems and Signal Processing
A well-conditioned estimator for large-dimensional covariance matrices
Journal of Multivariate Analysis
Journal of Multivariate Analysis
Journal of Multivariate Analysis
Complex singular wishart matrices and applications
Computers & Mathematics with Applications
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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 samples. The Stein-Haff identities and calculus on eigenstructure for singular Wishart matrices are developed for real and complex cases, respectively. By using these techniques, the unbiased risk estimates for certain classes of estimators for the population covariance matrices under 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 [L.R. Haff, Empirical Bayes estimation of the multivariate normal covariance matrix, Ann. Statist. 8 (1980) 586-697] are shown to improve upon the best scalar multiple of the empirical covariance matrix under the invariant quadratic loss functions for both real and complex multivariate normal distributions in the situation where the dimension of the variables is larger than the number of samples.