## The Annals of Statistics

- Ann. Statist.
- Volume 19, Number 2 (1991), 952-960.

### Shrinkage Domination in a Multivariate Common Mean Problem

#### Abstract

Consider the problem of estimating the $p \times 1$ mean vector $\theta$ under expected squared error loss, based on the observation of two independent multivariate normal vectors $Y_1 \sim N_p(\theta, \sigma^2I)$ and $Y_2 \sim N_p(\theta, \lambda\sigma^2I)$ when $\lambda$ and $\sigma^2$ are unknown. For $p \geq 3$, estimators of the form $\delta_\eta = \eta Y_1 + (1 - \eta)Y_2$ where $\eta$ is a fixed number in (0, 1), are shown to be uniformly dominated in risk by Stein estimators in spite of the fact that independent estimates of scale are unavailable. A consequence of this result is that when $\lambda$ is assumed known, shrinkage domination is robust to incorrect specification of $\lambda$.

#### Article information

**Source**

Ann. Statist., Volume 19, Number 2 (1991), 952-960.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176348130

**Digital Object Identifier**

doi:10.1214/aos/1176348130

**Mathematical Reviews number (MathSciNet)**

MR1105854

**Zentralblatt MATH identifier**

0725.62051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62H12: Estimation

Secondary: 62C99: None of the above, but in this section 62J07: Ridge regression; shrinkage estimators

**Keywords**

Risk robustness shrinkage estimation Stein estimators

#### Citation

George, Edward I. Shrinkage Domination in a Multivariate Common Mean Problem. Ann. Statist. 19 (1991), no. 2, 952--960. doi:10.1214/aos/1176348130. https://projecteuclid.org/euclid.aos/1176348130