TY - JOUR

T1 - Estimating a multivariate normal mean with a bounded signal to noise ratio under scaled squared error loss

AU - Kortbi, Othmane

AU - Marchand, Éric

N1 - Funding Information:
Acknowledgement. The authors are grateful to two reviewers, an associate editor, and the editor for useful comments and suggestions which led to a more accessible manuscript. The research work of Éric Marchand is partially supported by NSERC of Canada. During Othmane Kortbi’s Ph.D. studies at the Universitéde Sherbrooke, he benefited from financial support from several sources but he wishes to thank especially the ISM (Institut de sciences mathématiques) and the CRM (Centre de recherches mathématiques). Finally, the authors are grateful to Bill Strawderman and Dominique Fourdrinier for useful discussions and encouraging us to pursue work on this problem.
Publisher Copyright:
© 2013, Indian Statistical Institute.

PY - 2013/8/1

Y1 - 2013/8/1

N2 - For normal models with X ~ Np(θ, σ2 Ip), 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 δUB(X) = X, or the maximum likelihood estimator δML(X,S2), 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 ≤ √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 ≤ √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.

AB - For normal models with X ~ Np(θ, σ2 Ip), 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 δUB(X) = X, or the maximum likelihood estimator δML(X,S2), 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 ≤ √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 ≤ √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.

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U2 - 10.1007/s13171-013-0028-x

DO - 10.1007/s13171-013-0028-x

M3 - Article

AN - SCOPUS:85034599205

VL - 75

SP - 277

EP - 299

JO - Sankhya A

JF - Sankhya A

SN - 0976-836X

IS - 2

ER -