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

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  • Fourdrinier, Dominique
  • Marchand, Éric
  • Strawderman, William E.

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}.

Suggested Citation

  • Fourdrinier, Dominique & Marchand, Éric & Strawderman, William E., 2019. "On efficient prediction and predictive density estimation for normal and spherically symmetric models," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 18-25.
  • Handle: RePEc:eee:jmvana:v:173:y:2019:i:c:p:18-25
    DOI: 10.1016/j.jmva.2019.02.002
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    References listed on IDEAS

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    1. Kengo Kato, 2009. "Improved prediction for a multivariate normal distribution with unknown mean and variance," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(3), pages 531-542, September.
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    4. Mohammad Jafari Jozani & Éric Marchand & William Strawderman, 2014. "Estimation of a non-negative location parameter with unknown scale," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(4), pages 811-832, August.
    5. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E., 2015. "On improved shrinkage estimators for concave loss," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 241-246.
    6. Ann Brandwein & Stefan Ralescu & William Strawderman, 1993. "Shrinkage estimators of the location parameter for certain spherically symmetric distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(3), pages 551-565, September.
    7. A. Boisbunon & Y. Maruyama, 2014. "Inadmissibility of the best equivariant predictive density in the unknown variance case," Biometrika, Biometrika Trust, vol. 101(3), pages 733-740.
    8. Fourdrinier, Dominique & Strawderman, William E. & Wells, Martin T., 2003. "Robust shrinkage estimation for elliptically symmetric distributions with unknown covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 85(1), pages 24-39, April.
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