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

Othmane Kortbi, Éric Marchand

Research output: Contribution to journalArticlepeer-review


For normal models with X ~ Np(θ,σ2 Ip), S ~ σ2χ2k, independent, we consider the problem of estimating θ under scale invariant squared error loss ||d-θ||22, 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, S), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δbu,o associated with a prior on (θ,σ) such that θ\σ is uniformly distributed on the (boundary) sphere of radius mσ and a non-informative 1/σ prior measure is placed marginally on σ. With a series of technical results related to δbu,o; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever m < yfp and p > 2, δbu,o dominates both δub and δml. The finding can be viewed as both a multivariate extension of p = 1 result due to Kubokawa (Sankhy-a 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 < y/p/2, a wide class of Bayes estimators, which include priors where θ\σ is uniformly distributed on the ball of radius mσ centered at the origin, are shown to dominate δub.

Original languageEnglish
Pages (from-to)277-299
Number of pages23
JournalSankhya: The Indian Journal of Statistics
Volume75 A
Issue numberPART2
Publication statusPublished - 2013
Externally publishedYes


  • Bayes estimators
  • Coefficient of variation
  • Confluent hypergeometric functions
  • Dominance
  • Estimation
  • Maximum likelihood
  • Multivariate normal
  • Restricted parameter
  • Signal-to-noissquared error loss

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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