## The Annals of Statistics

### Estimating the Common Mean of Two Multivariate Normal Distributions

Wei-Liem Loh

#### Abstract

Let $X_1, X_2$ be two $p \times 1$ multivariate normal random vectors and $S_1, S_2$ be two $p \times p$ Wishart matrices, where $X_1 \sim N_p(\xi, \sum_1), X_2 \sim N_p(\xi, \sum_2), S_1 \sim W_p(\sum_1, n)$ and $S_2 \sim W_p(\sum_2, n)$. We further assume that $X_1, X_2, S_1, S_2$ are stochastically independent. We wish to estimate the common mean $\xi$ with respect to the loss function $L = (\hat{\xi} - \xi)'(\sum^{-1}_1 + \sum^{-1}_2)(\hat{\xi} - \xi)$. By extending the methods of Stein and Haff, an alternative unbiased estimator to the usual generalized least squares estimator is obtained. However, the risk of this estimator is not available in closed form. A Monte Carlo swindle is used instead to evaluate its risk performance. The results indicate that the alternative estimator performs very favorably against the usual estimator.

#### Article information

Source
Ann. Statist., Volume 19, Number 1 (1991), 297-313.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347983

Digital Object Identifier
doi:10.1214/aos/1176347983

Mathematical Reviews number (MathSciNet)
MR1091852

Zentralblatt MATH identifier
0742.62055

JSTOR