The Annals of Statistics

Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution

Bradley Efron and Carl Morris

Full-text: Open access


Ever since Stein's result, that the sample mean vector $\mathbf{X}$ of a $k \geqq 3$ dimensional normal distribution is an inadmissible estimator of its expectation $\mathbf{\theta}$, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of $\mathbf{\theta}$ when the dispersion matrix $\sum$ of $\mathbf{X}$ is unknown and must be estimated. Stein obtained this result earlier for known $\mathbf{\sum}$. Minimax conditions which are weaker than any yet published are derived by finding all estimators whose unbiased estimate of risk is bounded uniformly by $k$, the risk of $\mathbf{X}$. One sequence of risk functions and risk estimates applies simultaneously to the various assumptions about $\mathbf{\sum}$, resulting in a unified theory for these situations.

Article information

Ann. Statist., Volume 4, Number 1 (1976), 11-21.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62C99: None of the above, but in this section

Estimation minimax estimators risk of invariant estimators mean of a multivariate normal distribution Stein's estimator


Efron, Bradley; Morris, Carl. Families of Minimax Estimators of the Mean of a Multivariate Normal Distribution. Ann. Statist. 4 (1976), no. 1, 11--21. doi:10.1214/aos/1176343344.

Export citation