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

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 }.