Estimation of a multivariate normal mean with a bounded signal to noise ratio

Abstract

For normal canonical models with $X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;{independent}$, we consider the problem of estimating $\theta$ under scale invariant squared error loss $\frac{\|d-\theta \|^{2}}{\sigma^{2}}$, when it is known that the signal-to-noise ratio $\frac{\|\theta\|}{\sigma}$ is bounded above by $m$. Risk analysis is achieved by making use of a conditional risk decomposition and we obtain in particular sufficient conditions for an estimator to dominate either the unbiased estimator $\delta_{UB}(X)=X$, or the maximum likelihood estimator $\delta_{\hbox{mle}}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator $\delta_{BU}$ associated with a prior on $(\theta,\sigma)$ such that $\theta|\sigma$ is uniformly distributed on the (boundary) sphere of radius $m$ and a non-informative $\frac{1}{\sigma}$ prior measure is placed marginally on $\sigma$. With a series of technical results related to $\delta_{BU}$; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m \leq \sqrt{p}$ and $p \geq 2$, $\delta_{BU}$ dominates both $\delta_{UB}$ and $\delta_{\hbox{mle}}$. The finding can be viewed as both a multivariate extension of $p=1$ result due to Kubokawa (2005) and a unknown variance extension of a similar dominance finding due to Marchand and Perron (2001). Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for $m \leq \sqrt{\frac{p}{2}}$, a wide class of Bayes estimators, which include priors where $\theta|\sigma$ is uniformly distributed on the ball of radius $m$, are shown to dominate $\delta_{UB}$.