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

Dominique Fourdrinier, Éric Marchand and William E. Strawderman

Journal of Multivariate Analysis, 2019, vol. 173, issue C, 18-25

Abstract: 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}.

Keywords: Bayes estimator; Dominance; Duality; Kullback–Leibler; Multivariate normal; Multivariate student; Plug-in; Point prediction; Predictive densities; Scale mixture of normals; Spherically symmetric (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0047259X18304755
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:jmvana:v:173:y:2019:i:c:p:18-25

Ordering information: This journal article can be ordered from
http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.jmva.2019.02.002

Access Statistics for this article

Journal of Multivariate Analysis is currently edited by de Leeuw, J.

More articles in Journal of Multivariate Analysis from Elsevier
Bibliographic data for series maintained by Catherine Liu ().