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), S ~ σ²χ²k, 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 δml (X, S), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δbu,o associated with a prior on (θ,σ) such that θ\σ is uniformly distributed on the (boundary) sphere of radius mσ and a non-informative 1/σ prior measure is placed marginally on σ. With a series of technical results related to δbu,o; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever m < yfp and p > 2, δbu,o dominates both δub and δml. The finding can be viewed as both a multivariate extension of p = 1 result due to Kubokawa (Sankhy-a 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 < y/p/2, a wide class of Bayes estimators, which include priors where θ\σ is uniformly distributed on the ball of radius mσ centered at the origin, are shown to dominate δub.