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- W1569128504 abstract "For normal canonical models with $X sim N_p(theta, sigma^{2} I_{p}), ;; S^{2} sim sigma^{2}chi^{2}_{k}, ;{independent}$, we consider the problem of estimating $theta$ under scale invariant squared error loss $frac{|d-theta |^{2}}{sigma^{2}}$, when it is known that the signal-to-noise ratio $frac{|theta|}{sigma}$ 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 $delta_{UB}(X)=X$, or the maximum likelihood estimator $delta_{hbox{mle}}(X,S^2)$, or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator $delta_{BU}$ associated with a prior on $(theta,sigma)$ such that $theta|sigma$ is uniformly distributed on the (boundary) sphere of radius $m$ and a non-informative $frac{1}{sigma}$ prior measure is placed marginally on $sigma$. With a series of technical results related to $delta_{BU}$; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever $m leq sqrt{p}$ and $p geq 2$, $delta_{BU}$ dominates both $delta_{UB}$ and $delta_{hbox{mle}}$. 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 $m leq sqrt{frac{p}{2}}$, a wide class of Bayes estimators, which include priors where $theta|sigma$ is uniformly distributed on the ball of radius $m$, are shown to dominate $delta_{UB}$." @default.
- W1569128504 created "2016-06-24" @default.
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- W1569128504 date "2012-04-26" @default.
- W1569128504 modified "2023-09-27" @default.
- W1569128504 title "Estimation of a multivariate normal mean with a bounded signal to noise ratio" @default.
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