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

Abstract

For normal canonical models with X ∼ N p (θ, σ 2 I p ), S 2 ∼ σ 2 χ 2 k , independent, we consider the problem of estimating θ under scale invariant squared error loss d−θ 2 σ 2 , 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 U B (X) = X, or the maximum likelihood estimator δ 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 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 ≤ √ p and p ≥ 2, δ BU dominates both δ U B and δ 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 ≤ 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 U B .