期刊名称：Sankhya. Series A, mathematical statistics and probability
印刷版ISSN：0976-836X
电子版ISSN：0976-8378
出版年度：2013
卷号：75
期号：2
页码：277-299
DOI：10.1007/s13171-013-0028-x
语种：English
出版社：Indian Statistical Institute
摘要：For normal models with \(X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;\mbox{independent}\) , 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 \({\left\|\theta\right\|}/{\sigma}\) 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 \(\delta_{ML}(X,S^2)\) , or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δ BU ,0 associated with a prior on ( θ , σ 2) such that θ | σ 2 is uniformly distributed on the (boundary) sphere of radius m σ and a non-informative 1/ σ 2 prior measure is placed marginally on σ 2. With a series of technical results related to δ BU ,0; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever \(m \leq \sqrt{p}\) 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 \(m \leq \sqrt{p/2}\) , a wide class of Bayes estimators, which include priors where θ | σ 2 is uniformly distributed on the ball of radius m σ centered at the origin, are shown to dominate δ UB .
关键词：Bayes estimators ; coefficient of variation ; confluent hypergeometric functions ; dominance ; estimation ; maximum likelihood ; multivariate normal ; restricted parameter ; signal-to-noise ratio ; squared error loss
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