## Abstract

For normal 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_{UB}(X) = X, or the maximum likelihood estimator δ_{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 mσ 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 ≤ √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 ≤ √p/2, a wide class of Bayes estimators, which include priors where θ|σ^{2} is uniformly distributed on the ball of radius mσ centered at the origin, are shown to dominate δ_{UB}.

Original language | English |
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Pages (from-to) | 277-299 |

Number of pages | 23 |

Journal | Sankhya A |

Volume | 75 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 1 2013 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty