For Wishart density functions, the risk dominance problems of moment estimators, maximum likelihood estimators (MLEs), James-Stein type minimax estimators and their improved estimators of covariance matrices under the Kullback-Leibler loss function have been well studied in the literature. However, attentions have mostly paid on the onesample and the two-sample problems. In this paper, it emphasizes on two-folds for the k-sample problem: (i) based on the sample covariance matrices (moment estimators), by incorporating the Cholesky decomposition to establish the James-Stein type estimators both over the whole parameter space and over the restricted parameter space under the simple tree ordering set, respectively, (ii) based on the corresponding MLEs, by incorporating the spectrum decomposition and its new generalization to propose a unified procedure to construct the improved estimators both over the whole parameter space and over the restricted parameter space under the simple tree ordering set, respectively. The results are directly applied to the estimation of covariance matrices for the completely balanced multivariate multi-way random effects models without interactions.
