TY - JOUR
T1 - On efficient prediction and predictive density estimation for normal and spherically symmetric models
AU - Fourdrinier, Dominique
AU - Marchand, Éric
AU - Strawderman, William E.
N1 - Funding Information: We are grateful to two reviewers for constructive comments and sharp corrections. Éric Marchand research is supported in part by the Natural Sciences and Engineering Research Council of Canada , and William E. Strawderman research is partially supported by grants from the Simons Foundation, USA ( #209035 and #418098 ).
PY - 2019/9
Y1 - 2019/9
N2 - 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 }.
AB - 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 }.
KW - Bayes estimator
KW - Dominance
KW - Duality
KW - Kullback–Leibler
KW - Multivariate normal
KW - Multivariate student
KW - Plug-in
KW - Point prediction
KW - Predictive densities
KW - Scale mixture of normals
KW - Spherically symmetric
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U2 - https://doi.org/10.1016/j.jmva.2019.02.002
DO - https://doi.org/10.1016/j.jmva.2019.02.002
M3 - Article
VL - 173
SP - 18
EP - 25
JO - Journal of Multivariate Analysis
JF - Journal of Multivariate Analysis
SN - 0047-259X
ER -