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

Abstract

Abstract Let X , Y , U be independent distributed as X ∼ N d ( θ , σ 2 I d ) , Y ∼ N d ( c θ , 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 − c θ ‖ 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 q ˆ ( ⋅ ; x , u ) for Y as measured by the expected Kullback–Leibler loss. A benchmark procedure is the minimum risk equivariant density q ˆ MRE , which is generalized Bayes with respect to the prior π ( θ , η ) = 1 ∕ η . In dimension d ≥ 3 , we obtain improvements on q ˆ 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 q ˆ 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 c X simultaneously for all f , and simultaneously for risk functions E E 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 } .