For a multivariate normal set-up, it is well known that themaximumlikelihood estimator (MLE) of covariance matrix is neither admissible nor minimax under the Stein loss function. In this paper, we reveal that the MLE based on the Iwasawa parameterization leads to minimaxity with respect to the Stein loss function. Furthermore, a novel class of loss functions is proposed so that the minimum risks of the MLEs are identical in different coordinate systems, Cholesky parameterization and full Iwasawa parameterization. In other words, the MLEs based on these two different parameterizations are characterized by the property of minimaxity, without a Stein paradox. The application of our novel method to the high-dimensional covariance matrix problem is also discussed.

中文翻译：

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关于协方差矩阵的最大似然估计

对于多元正态设置，众所周知，在Stein损失函数下，协方差矩阵的最大似然估计器（MLE）既不可接受也不是minimax。在本文中，我们揭示了基于Iwasawa参数化的MLE导致关于Stein损失函数的最小化。此外，提出了一类新的损失函数，以使MLE的最小风险在不同的坐标系，Cholesky参数化和完全Iwasawa参数化中相同。换句话说，基于这两个不同参数化的MLE的特征在于最小化特性，而没有Stein悖论。还讨论了我们的新方法在高维协方差矩阵问题中的应用。