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 - 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
SN - 0976-836X
VL - 75
SP - 277
EP - 299
JO - Sankhya A
JF - Sankhya A
IS - 2
ER -