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 $\lambda$.