Title: A note on MLE of covariance matrix,
Abstract: For a multivariate normal set up, it is well known that the maximum
likelihood estimator of covariance matrix is neither admissible nor minimax
under the Stein loss function. For the past six decades, a bunch of researches
have followed along this line for Stein's phenomenon in the literature. In this
note, the results are two folds: Firstly, with respect to Stein type loss
function we use the full Iwasawa decomposition to enhance the unpleasant
phenomenon that the minimum risks of maximum likelihood estimators for the
different coordinate systems (Cholesky decomposition and full Iwasawa
decomposition) are different. Secondly, we introduce a new class of loss
functions to show that the minimum risks of maximum likelihood estimators for
the different coordinate systems, the Cholesky decomposition and the full
Iwasawa decomposition, are of the same, and hence the Stein's paradox
disappears.