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Shrinkage Estimators under Spherical Symmetry for the General Linear Model

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  • Cellier, D.
  • Fourdrinier, D.

Abstract

This paper is primarily concerned with extending the results of Brandwein and Strawderman in the usual canonical setting of a general linear model when sampling from a spherically symmetric distribution. When the location parameter belongs to a proper linear subspace of the sampling space, we give an unbiased estimator of the difference of the risks between the least squares estimator [phi]0 and a general shrinkage estimator [phi] = [phi]0 - [short parallel]X - [phi]0 [short parallel]2 · g o [phi]0. We obtain a general condition of domination for [phi] over [phi]0 which is weaker than that of Brandwein and Strawderman. We do not need any superharmonicity condition on g. Our results are valid for general quadratic loss.

Suggested Citation

  • Cellier, D. & Fourdrinier, D., 1995. "Shrinkage Estimators under Spherical Symmetry for the General Linear Model," Journal of Multivariate Analysis, Elsevier, vol. 52(2), pages 338-351, February.
    • Handle: RePEc:eee:jmvana:v:52:y:1995:i:2:p:338-351
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    Cited by:

    1. Marchand, Éric & Perron, François, 2005. "Improving on the mle of a bounded location parameter for spherical distributions," Journal of Multivariate Analysis, Elsevier, vol. 92(2), pages 227-238, February.
    2. Fourdrinier, Dominique & Strawderman, William E., 2008. "A unified and generalized set of shrinkage bounds on minimax Stein estimates," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2221-2233, November.
    3. 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.
    4. Kazuhiro Ohtani, 1998. "An MSE comparison of the restricted Stein-rule and minimum mean squared error estimators in regression," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 361-376, December.
    5. Dominique Fourdrinier & William Strawderman, 2015. "Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(4), pages 461-484, May.
    6. Jurečková Jana & Sen P. K., 2006. "Robust multivariate location estimation, admissibility, and shrinkage phenomenon," Statistics & Risk Modeling, De Gruyter, vol. 24(2), pages 273-290, December.
    7. Kubokawa, T. & Srivastava, M. S., 2001. "Robust Improvement in Estimation of a Mean Matrix in an Elliptically Contoured Distribution," Journal of Multivariate Analysis, Elsevier, vol. 76(1), pages 138-152, January.
    8. Ouassou, Idir & Strawderman, William E., 2002. "Estimation of a parameter vector restricted to a cone," Statistics & Probability Letters, Elsevier, vol. 56(2), pages 121-129, January.
    9. Fourdrinier, Dominique & Ouassou, Idir & Strawderman, William E., 2003. "Estimation of a parameter vector when some components are restricted," Journal of Multivariate Analysis, Elsevier, vol. 86(1), pages 14-27, July.
    10. Aurélie Boisbunon & Stéphane Canu & Dominique Fourdrinier & William Strawderman & Martin T. Wells, 2014. "Akaike's Information Criterion, C p and Estimators of Loss for Elliptically Symmetric Distributions," International Statistical Review, International Statistical Institute, vol. 82(3), pages 422-439, December.
    11. Dominique Fourdrinier & Tatsuya Kubokawa & William E. Strawderman, 2023. "Shrinkage Estimation of a Location Parameter for a Multivariate Skew Elliptic Distribution," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 808-828, February.
    12. Dey, Dipak K. & Ghosh, Malay & Strawderman, William E., 1999. "On estimation with balanced loss functions," Statistics & Probability Letters, Elsevier, vol. 45(2), pages 97-101, November.
    13. He Kun & Strawderman William E., 2001. "Estimation In Spherically Symmetric Regression With Random Design," Statistics & Risk Modeling, De Gruyter, vol. 19(1), pages 41-50, January.
    14. Fourdrinier Dominique & Strawderman William E. & Wells Martin T., 2009. "Improved estimation for elliptically symmetric distributions with unknown block diagonal covariance matrix," Statistics & Risk Modeling, De Gruyter, vol. 26(3), pages 203-217, April.
    15. Fourdrinier Dominique & Lemaire Anne-Sophie, 2000. "ESTIMATION OF THE MEAN OF A e1-EXPONENTIAL MULTIVARIATE DISTRIBUTION," Statistics & Risk Modeling, De Gruyter, vol. 18(3), pages 259-274, March.

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