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

For normal canonical models with $X \sim N_p(θ, σ^{2} I_{p}), \;\; S^{2} \sim σ^{2}χ^{2}_{k}, \;{independent}$, we consider the problem of estimating $θ$ under scale invariant squared error loss $\frac{\|d-θ\|^{2}}{σ^{2}}$, when it is known that the signal-to-noise ratio $\frac{\|θ\|}σ$ 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 $δ_{UB}(X)=X$, or the maximum likelihood estimator $δ_{\hbox{mle}}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator $δ_{BU}$ associated with a prior on $(θ,σ)$ such that $θ|σ$ is uniformly distributed on the (boundary) sphere of radius $m$ and a non-informative $\frac{1}σ$ prior measure is placed marginally on $σ$. With a series of technical results related to $δ_{BU}$; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m \leq \sqrt{p}$ and $p \geq 2$, $δ_{BU}$ dominates both $δ_{UB}$ and $δ_{\hbox{mle}}$. 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 $θ|σ$ is uniformly distributed on the ball of radius $m$, are shown to dominate $δ_{UB}$.

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