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Robust minimax Stein estimation under invariant data-based loss for spherically and elliptically symmetric distributions

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  • Dominique Fourdrinier
  • William Strawderman

Abstract

From an observable $$(X,U)$$ ( X , U ) in $$\mathbb R^p \times \mathbb R^k$$ R p × R k , we consider estimation of an unknown location parameter $$\theta \in \mathbb R^p$$ θ ∈ R p under two distributional settings: the density of $$(X,U)$$ ( X , U ) is spherically symmetric with an unknown scale parameter $$\sigma $$ σ and is ellipically symmetric with an unknown covariance matrix $$\Sigma $$ Σ . Evaluation of estimators of $$\theta $$ θ is made under the classical invariant losses $$\Vert d - \theta \Vert ^2 / \sigma ^2$$ ‖ d - θ ‖ 2 / σ 2 and $$(d - \theta )^t \Sigma ^{-1} (d - \theta )$$ ( d - θ ) t Σ - 1 ( d - θ ) as well as two respective data based losses $$\Vert d - \theta \Vert ^2 / \Vert U\Vert ^2$$ ‖ d - θ ‖ 2 / ‖ U ‖ 2 and $$(d - \theta )^t S^{-1} (d - \theta )$$ ( d - θ ) t S - 1 ( d - θ ) where $$\Vert U\Vert ^2$$ ‖ U ‖ 2 estimates $$\sigma ^2$$ σ 2 while $$S$$ S estimates $$\Sigma $$ Σ . We provide new Stein and Stein–Haff identities that allow analysis of risk for these two new losses, including a new identity that gives rise to unbiased estimates of risk (up to a multiple of $$1 / \sigma ^2$$ 1 / σ 2 ) in the spherical case for a larger class of estimators than in Fourdrinier et al. (J Multivar Anal 85:24–39, 2003 ). Minimax estimators of Baranchik form illustrate the theory. It is found that the range of shrinkage of these estimators is slightly larger for the data based losses compared to the usual invariant losses. It is also found that $$X$$ X is minimax with finite risk with respect to the data-based losses for many distributions for which its risk is infinite when calculated under the classical invariant losses. In these cases, including the multivariate $$t$$ t and, in particular, the multivariate Cauchy, we find improved shrinkage estimators as well. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
  • Handle: RePEc:spr:metrik:v:78:y:2015:i:4:p:461-484
    DOI: 10.1007/s00184-014-0512-x
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    References listed on IDEAS

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    1. Fourdrinier, Dominique & Strawderman, William E., 1996. "A Paradox Concerning Shrinkage Estimators: Should a Known Scale Parameter Be Replaced by an Estimated Value in the Shrinkage Factor?," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 109-140, November.
    2. Haff, L. R., 1979. "An identity for the Wishart distribution with applications," Journal of Multivariate Analysis, Elsevier, vol. 9(4), pages 531-544, December.
    3. 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.
    4. Fourdrinier, Dominique & Strawderman, William, 2014. "On the non existence of unbiased estimators of risk for spherically symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 6-13.
    5. Fourdrinier, D. & Wells, M. T., 1995. "Loss Estimation for Spherically Symmetrical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 53(2), pages 311-331, May.
    6. Cellier, Dominique & Fourdrinier, Dominique & Robert, Christian, 1989. "Robust shrinkage estimators of the location parameter for elliptically symmetric distributions," Journal of Multivariate Analysis, Elsevier, vol. 29(1), pages 39-52, April.
    7. T. Kubokawa & C. Robert & A. Saleh, 1991. "Robust estimation of common regression coefficients under spherical symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 43(4), pages 677-688, December.
    8. Dominique Fourdrinier & William Strawderman & Martin Wells, 2006. "Estimation of a Location Parameter with Restrictions or “vague information” for Spherically Symmetric Distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 58(1), pages 73-92, March.
    9. 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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    Cited by:

    1. Ruili Sun & Tiefeng Ma & Shuangzhe Liu, 2018. "A Stein-type shrinkage estimator of the covariance matrix for portfolio selections," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(8), pages 931-952, November.
    2. Fourdrinier, Dominique & Haddouche, Anis M. & Mezoued, Fatiha, 2021. "Covariance matrix estimation under data-based loss," Statistics & Probability Letters, Elsevier, vol. 177(C).
    3. Stéphane Canu & Dominique Fourdrinier, 2023. "Data based loss estimation of the mean of a spherical distribution with a residual vector," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(8), pages 851-878, November.
    4. Hamid Karamikabir & Nasrin Karamikabir & Mohammad Ali Khajeian & Mahmoud Afshari, 2023. "Bayesian Wavelet Stein’s Unbiased Risk Estimation of Multivariate Normal Distribution Under Reflected Normal Loss," Methodology and Computing in Applied Probability, Springer, vol. 25(1), pages 1-20, March.
    5. Karamikabir, Hamid & Afshari, Mahmoud, 2020. "Generalized Bayesian shrinkage and wavelet estimation of location parameter for spherical distribution under balance-type loss: Minimaxity and admissibility," Journal of Multivariate Analysis, Elsevier, vol. 177(C).

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