Estimating a multivariate normal mean with a bounded signal to noise ratio under scaled squared error loss

Abstract

For normal models with \(X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;\mbox{independent}\), we consider the problem of estimating θ under scale invariant squared error loss ||d − θ||2/σ2, when it is known that the signal-to-noise ratio \({\left\|\theta\right\|}/{\sigma}\) 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 δUB(X) = X, or the maximum likelihood estimator \(\delta_{ML}(X,S^2)\), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δBU,0 associated with a prior on (θ,σ2) such that θ|σ2 is uniformly distributed on the (boundary) sphere of radius mσ and a non-informative 1/σ2 prior measure is placed marginally on σ2. With a series of technical results related to δBU,0; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever \(m \leq \sqrt{p}\) and p ≥ 2, δBU,0 dominates both δUB and δML. The finding can be viewed as both a multivariate extension of p = 1 result due to Kubokawa (Sankhyā 67:499–525, 2005) and an unknown variance extension of a similar dominance finding due to Marchand and Perron (Ann Stat 29:1078–1093, 2001). Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for \(m \leq \sqrt{p/2}\), a wide class of Bayes estimators, which include priors where θ|σ2 is uniformly distributed on the ball of radius mσ centered at the origin, are shown to dominate δUB.