On efficient prediction and predictive density estimation for normal and spherically symmetric models

Dominique Fourdrinier, Éric Marchand, William E. Strawderman

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


Let X,Y,U be independent distributed as X∼N d (θ,σ 2 I d ), Y∼N d (cθ,σ 2 I d ), and U U∼σ 2 χ k 2 , or more generally spherically symmetric distributed with density η d+k∕2 f{η(‖x−θ‖ 2 +‖u‖ 2 +‖y−cθ‖ 2 )}, with unknown parameters θ∈R d and η=1∕σ 2 >0, known density f, and c∈R + . Based on observing X=x,U=u, we consider the problem of obtaining a predictive density qˆ(⋅;x,u) for Y as measured by the expected Kullback–Leibler loss. A benchmark procedure is the minimum risk equivariant density qˆ MRE , which is generalized Bayes with respect to the prior π(θ,η)=1∕η. In dimension d≥3, we obtain improvements on qˆ MRE , and further show that the dominance holds simultaneously for all f subject to finite moment and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior π h (θ,η)=‖θ‖ 2−d ∕η dominates qˆ MRE simultaneously for all scale mixture of normals f. The results hinge on duality with a point prediction problem, as well as posterior representations for (θ,η), which are very much of interest on their own. Namely, we obtain for d≥3, point predictors δ(X,U) of Y that dominate the benchmark predictor cX simultaneously for all f, and simultaneously for risk functions EE f [ρ{‖Y−δ(X,U)‖ 2 +(1+c 2 )‖U‖ 2 }], with ρ increasing and concave on R + , and including the squared error case E f {‖Y−δ(X,U)‖ 2 }.

Original languageEnglish (US)
Pages (from-to)18-25
Number of pages8
JournalJournal of Multivariate Analysis
StatePublished - Sep 2019

ASJC Scopus subject areas

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty


  • Bayes estimator
  • Dominance
  • Duality
  • Kullback–Leibler
  • Multivariate normal
  • Multivariate student
  • Plug-in
  • Point prediction
  • Predictive densities
  • Scale mixture of normals
  • Spherically symmetric


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