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

Let X,Y,U be independent distributed as X∼Nd(θ,σ2Id), Y∼Nd(cθ,σ2Id), and U⊤U∼σ2χk2, or more generally spherically symmetric distributed with density ηd+k∕2f{η(‖x−θ‖2+‖u‖2+‖y−cθ‖2)}, with unknown parameters θ∈Rd and η=1∕σ2>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 qˆMRE, which is generalized Bayes with respect to the prior π(θ,η)=1∕η. In dimension d≥3, we obtain improvements on qˆMRE, 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 πh(θ,η)=‖θ‖2−d∕η dominates qˆMRE 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 EEf[ρ{‖Y−δ(X,U)‖2+(1+c2)‖U‖2}], with ρ increasing and concave on R+, and including the squared error case Ef{‖Y−δ(X,U)‖2}.