## The Annals of Statistics

### Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix

#### Abstract

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ based on $\mathbf{X}$ and $\mathbf{A}$ through $\mathbf{X}$ and $\mathbf{X}'\mathbf{A}^{-1}\mathbf{X}$. It is shown that there are minimax estimators of the form $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$ which are also generalized Bayes. A special case where $\Sigma = \sigma^2\mathbf{I}$ is also considered.

#### Article information

Source
Ann. Statist., Volume 1, Number 1 (1973), 142-145.

Dates
First available in Project Euclid: 25 October 2007

https://projecteuclid.org/euclid.aos/1193342390

Digital Object Identifier
doi:10.1214/aos/1193342390

Mathematical Reviews number (MathSciNet)
MR331573

Zentralblatt MATH identifier
0254.62006

#### Citation

Lin, Pi-Erh; Tsai, Hui-Liang. Generalized Bayes Minimax Estimators of the Multivariate Normal Mean with Unknown Covariance Matrix. Ann. Statist. 1 (1973), no. 1, 142--145. doi:10.1214/aos/1193342390. https://projecteuclid.org/euclid.aos/1193342390