A Note on Improved Multivariate Normal Mean Estimation With Unknown Covariance When p Is Greater Than n

TL;DR

This paper identifies and corrects critical errors in the proofs of key results in a previous study on multivariate normal mean estimation with unknown covariance when the dimension exceeds the sample size.

Contribution

It revises the proof of a significant theorem under realistic assumptions about the covariance estimator's rank, ensuring the validity of the earlier results.

Findings

Corrected proof of the main theorem.

Clarification of assumptions on covariance estimator.

Reinforcement of the original results' validity.

Abstract

In this paper, we highlight a major error in the proofs of the important results of [D.Ch\'etelat and M. T. Wells(2012). Improved Multivariate Normal Mean Estimation with Unknown Covariance when p is Greater than n. The Annals of Statistics, Vol. 40, No.6, 3137--3160]. In particular, the proofs of some of their main results are based on Theorem 2 whose proof needs to be revisited. More precisely, there are some major mistakes in the derivation of this important result. Further, under a very realistic assumption about the rank of the estimator of the variance-covariance matrix, we correct the proof of the quoted result.