# A note on MLE of covariance matrix

## TL;DR

This paper investigates the maximum likelihood estimator of covariance matrices in multivariate normal models, showing how different decompositions affect risk and proposing new loss functions to resolve Stein's paradox.

## Contribution

It introduces a new class of loss functions that equalize risks across different matrix decompositions, addressing longstanding issues in covariance estimation.

## Key findings

- Full Iwasawa decomposition reduces risk differences
- New loss functions eliminate Stein's paradox
- MLE risks are equalized across decompositions

## 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.