Maximum likelihood estimation of Wishart mean matrices under Löwner order restrictions

For Wishart density functions, there remains a long-time question unsolved. That is whether there exists the closed-form MLEs of mean matrices over the partially Löwner ordering sets. In this note, we provide an affirmative answer by demonstrating a unified procedure on exactly how the closed-form MLEs are obtained for the simple ordering case. Under the Kullback-Leibler loss function, a property of obtained MLEs is further studied. Some applications of the obtained closed-form MLEs, including the comparison between our ML estimates and Calvin and Dykstra's [Maximum likelihood estimation of a set of covariance matrices under Löwner order restrictions with applications to balanced multivariate variance components models, Ann. Statist. 19 (1991) 850-869.] which obtained by iterative algorithm, are also made.