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

For normal models with X ~ N_p(θ, σ² I_p), S² ~σ²χ² _k, 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 ||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_ML(X,S²), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δ_BU,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 δ_BU,0; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever m ≤ √p and p ≥ 2, δ_BU,0 dominates both δ_UB and δ_ML. 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 ≤ √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 δ_UB.