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On Predictive Density Estimation for Location Families under Integrated L 2 and L 1 Losses

Author

Listed:
  • Tatsuya Kubokawa

    (Faculty of Economics, The University of Tokyo)

  • Éric Marchand

    (Université de Sherbrooke, Departement de mathématiques)

  • William E. Strawderman

    (Rutgers University, Department of Statistics and Biostatistics,)

Abstract

Our investigation concerns the estimation of predictive densities and a study of effiency as measured by the frequentist risk of such predictive densities with integrated L2 and L1 losses. Our findings relate to a p-variate spherically symmetric observable X ∼ px (||x -μ||2) and the objective of estimating the density of Y ∼ qY (||y - μ||2) based on X. For L2 loss, we describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three of more dimensions (i.e., p ≥ 3), we show that the MRE estimator is inadmissible under L2 loss and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of ||δ - μ||2. We also study the phenomenon of improvement on the plug-in density estimator of the form qY (||y - aX ||2), 0 1, showing in some cases, inevitably for large enough p, that all choices c > 1 are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for p ≥ 3. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with p ≥ 4 under L1 loss.

Suggested Citation

  • Tatsuya Kubokawa & Éric Marchand & William E. Strawderman, 2014. "On Predictive Density Estimation for Location Families under Integrated L 2 and L 1 Losses," CIRJE F-Series CIRJE-F-935, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2014cf935
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    References listed on IDEAS

    as
    1. Éric Marchand & William Strawderman, 2005. "Improving on the minimum risk equivariant estimator of a location parameter which is constrained to an interval or a half-interval," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 57(1), pages 129-143, March.
    2. Hartigan, J. A., 2004. "Uniform priors on convex sets improve risk," Statistics & Probability Letters, Elsevier, vol. 67(4), pages 285-288, May.
    3. Strawderman, William E., 1974. "Minimax estimation of location parameters for certain spherically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 4(3), pages 255-264, September.
    4. J. F. Lawless & Marc Fredette, 2005. "Frequentist prediction intervals and predictive distributions," Biometrika, Biometrika Trust, vol. 92(3), pages 529-542, September.
    5. Berg, C. & Vignat, C., 2010. "On the density of the sum of two independent Student t-random vectors," Statistics & Probability Letters, Elsevier, vol. 80(13-14), pages 1043-1055, July.
    6. Nason, Guy P., 2006. "On the sum of t and Gaussian random variables," Statistics & Probability Letters, Elsevier, vol. 76(12), pages 1280-1286, July.
    7. 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.
    8. Kano, Y., 1994. "Consistency Property of Elliptic Probability Density Functions," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 139-147, October.
    9. Kubokawa, Tatsuya & Marchand, Éric & Strawderman, William E. & Turcotte, Jean-Philippe, 2013. "Minimaxity in predictive density estimation with parametric constraints," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 382-397.
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    Cited by:

    1. 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.

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