## The Annals of Statistics

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343344

**Digital Object Identifier**

doi:10.1214/aos/1176343344

**Mathematical Reviews number (MathSciNet)**

MR403001

**Zentralblatt MATH identifier**

0322.62010

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F10: Point estimation

Secondary: 62C99: None of the above, but in this section

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176343344